Coloring the Mu transpososome
© Darcy et al; licensee BioMed Central Ltd. 2006
Received: 30 November 2005
Accepted: 05 October 2006
Published: 05 October 2006
Tangle analysis has been applied successfully to study proteins which bind two segments of DNA and can knot and link circular DNA. We show how tangle analysis can be extended to model any stable protein-DNA complex.
We discuss a computational method for finding the topological conformation of DNA bound within a protein complex. We use an elementary invariant from knot theory called colorability to encode and search for possible DNA conformations. We apply this method to analyze the experimental results of Pathania, Jayaram, and Harshey (Cell 2002). We show that the only topological DNA conformation bound by Mu transposase which is biologically likely is the five crossing solution found by Pathania et al (although other possibilities are discussed).
Our algorithm can be used to analyze the results of the experimental technique described in Pathania et al in order to determine the topological conformation of DNA bound within a stable protein-DNA complex.
When modeling protein-DNA reactions, it is helpful to know how to draw the DNA segments bound by the protein. For example, does the DNA molecule cross itself within the protein complex or does it bend sharply? Tangle analysis can be used to determine the topological shape of the DNA segments bound by a protein complex. Tangle analysis does not determine the exact geometry and hence cannot determine the sharpness of DNA bending, but it can determine the overall topology. This can be used to infer which DNA sequences are likely to be close to each other within the protein-DNA complex  as well as other information useful for modeling protein-DNA reactions.
The focus of this paper is two-fold: (1) we describe a computational method for solving n-string tangle equations for small crossing solutions; (2) we apply this method to analyze the topology of the DNA bound in the Mu transpososome.
Although our current C++ program is specific for analyzing the results of , we would be happy to make any necessary modifications for solving any other system of tangle equations for small crossing solutions, especially those modeling experimental data. The source code is also available upon request.
The Mu transpososome is involved in DNA transposition. DNA transposition is the process by which a piece of DNA can change its location within a genome. The Mu transposition pathway involves the formation of a series of protein-DNA complexes (for more biology background, see [5, 6]). The Mu transpososome refers to the Mu transposase protein complex (Mu) and the three DNA segments bound by this protein complex. Since three DNA segments are bound by Mu, the Mu transpososome can be modeled by a 3-string tangle. An experimental technique called difference topology [7, 8, 5, 9, 10] combined with tangle analysis was used in  to determine that some of the Mu-DNA complexes can be modeled by the five crossing 3-string tangle shown in Fig. 1B. There are an infinite number of tangles that mathematically satisfy these experimental results (Darcy IK, Luecke J, Vazquez M: A tangle analysis of the Mu transpososome protein complex which binds three DNA segments, manuscript in preparation). These other conformations are very complicated and hence biologically unlikely to model the Mu transpososome, but they leave open the possibility that there are other biologically relevant models.
We describe a computational algorithm we have implemented which solves for biologically relevant topological conformations of DNA bound within the Mu transpososome using experimental results from . For the purposes of this paper, we will consider a solution to be biologically relevant if it has a 2-dimensional projection with at most eight crossings. Observe that the solution found in  has five crossings (Fig. 1B). Although we briefly describe in the Discussion and Conclusions section why we believe the Mu transpososome contains at most eight crossings, our main reason for choosing to limit solutions up to eight crossings is computational time. Currently our C++ algorithm takes two days on a Linux computer with AMD Opteron Processor (2.2 GHz cpu) to find solutions up through eight crossings. The speed of the algorithm can be significantly improved by, for example, parallelizing the algorithm and running it on a cluster. Hence the algorithm can be improved to find solutions with around ten crossings. But as the number of tangles grows exponentially with crossing number, it is unlikely that this method can be used to find solutions with more than fifteen crossings due to computation time.
The experimental technique used in  can be applied to any protein complex which stably binds two or more segments of DNA (see Discussion and Conclusions for limitations) in order to determine the topological conformation of the DNA bound by the protein complex. The results of such experiments can be analyzed using a modification of the software we developed for analyzing the Mu experiments. In other words, this software can be modified to solve any system of n-string tangle equations for solutions containing up to around ten crossings, including those modeling difference topology experiments applied to a protein complex that stably binds any number of segments of DNA.
An example of a tangle equation
Difference topology and tangle modeling
Observe that in Fig. 4 (bottom) the yellow and brown arrow heads in the Cre complex point in opposite directions. Based on the crystal structure of Cre complexed with DNA , it was assumed in  that the two Cre binding sites must be in anti-parallel orientation with respect to each other within the Cre-DNA complex. Note that for the configuration in Fig. 3, an even number of crossings between Cre binding sites are needed to achieve an anti-parallel orientation between the Cre binding when the Cre sites are directly repeated (Fig. 3A) while an odd number of crossings are necessary when the Cre binding sites are inversely repeated (Fig. 3B). In the Mu/Cre models in Fig. 5, sometimes an extra crossing not bound by either protein is needed for correct DNA orientation within the Cre protein complex, depending on the orientation of the Cre binding sites on the two loops. When comparing products where the Cre sites are placed on the same pair of loops but in different orientations, it was assumed that the extra crossing occurred with the higher crossing product. When this extra crossing exists, it is placed within the green annulus in our figures. Hence crossings within the green annulus, if any, represent crossings not trapped by either protein complex.
Determining the topological conformation of DNA bound by Mu is equivalent to solving the system of tangle equations in Fig. 6 for the 3-string tangle T. A solution is a topological approximation, given as a 2-dimensional projection of a 3-dimensional conformation.
An example of a 3-dimensional reconstruction using 2-dimensional tangle models is given in .
There exist four 3-branched supercoiled solutions to the tangle equations in Fig. 6. These solutions are shown in Fig. 7. They were obtained by solving a system of linear equations. For example, looking at the bottom left tangle equation in Fig. 6 in which the product is a right-handed three crossing knot, we observe that if the solution is a 3-branched supercoiled conformation with x crossings in one branch containing a Cre binding site and y crossings in the other branch containing the other Cre binding site, then x + y = 3 (compare to bottom left tangle equation in Fig. 5). If we let z be the number of crossings in the third branch, then the top left equation involving an unidentified four crossing link in Fig. 6 corresponds to the linear equation y + z = ± 4, while the equation involving the unidentified three crossing knot (middle left equation in Fig. 6) results in the equation x + z = ± 3. If we solve the system of linear equations, x + y = 3, y + z = ± 4, x + z = ± 3, we obtain x = 1, y = 2, z = 2 (Fig. 7A), x = 2, y = 1, z = -5 (Fig. 7B), x = -2, y = 5, z = -1 (Fig. 7C), x = 5, y = -2, z = -2 (Fig. 7D).
Note that we are actually solving four different systems of linear equations (where each system has a unique 3-branched supercoiled solution) depending on whether the top left four crossing link is right- or left-handed (y + z = ± 4) and whether the unidentified three crossing knot is right- or left-handed (x + z = ± 3).
The solutions shown in Fig. 7B, 7C, 7D contain more crossings than the solution in Fig. 7A. Also, the solutions in Fig. 7B, 7C, 7D contain left-handed crossings. As the substrate DNA was negatively supercoiled, one would expect a 3-branched supercoiled structure to contain right-handed twists, not left-handed twists. Hence the Fig. 7A solution  is biologically more likely than the other 3-branched supercoiled solutions (Also, the solutions in Fig. 7B, 7C, do not satisfy additional experiments in  not described here).
The solutions in Fig. 7 are the only solutions if one considers only 3-branched supercoiled DNA conformations, but the question remains whether there are any other biologically relevant solutions if we do not assume a 3-branched supercoiled structure. In the next section, we describe colorability, the tangle invariant which we use to search for solutions for T where the only restriction placed on T is that it has eight or fewer crossings. However, a thorough understanding of this invariant is not necessary to understand the main idea behind the algorithm discussed in Results.
The coloring invariants
A coloring matrix of a knot/link/tangle diagram, T, is any matrix, M T , which is row equivalent to a coefficient matrix corresponding to the coloring equations. For example, the 6 × 8 matrix in Eqn. (1) is a coloring matrix corresponding to the tangle diagram in Fig. 9C. Each row corresponds to one of the six crossings in the tangle diagram while each column represents one of the eight arcs, x5, x6, x7, x8, x1, x2, x3, x4 in the tangle diagram.
We will call the arcs which have one endpoint on the boundary of the tangle 3-ball endpoint arcs. The remaining arcs will be called interior arcs. Notice that we place the four columns corresponding to the endpoint arcs, x1, x2, x3, x4, as the four rightmost columns of the matrix M T . We can solve this system of equations by performing the following row operations: (1) exchange two rows (row i ↔ row j); (2) add a multiple of one row to a different row (row i → row i + t · row j, i ≠ j, t ∈ ); (3) multiply a row by -1 (row i ↔ -row i). Since we are working in m where to is an arbitrary integer, scaling a row is not allowed.
The first non-zero term in a row is called a leading entry. A matrix is in echelon form if (1) rows consisting of only zero's occur below rows containing at least one non-zero term; (2) each entry below a leading entry is zero; (3) If a ih and a jk are leading entries and if i <j, then h <k (i.e., the leading entries move to the right as the rows descend). An echelon form, EF(M T ) of the matrix in Eqn. (1) is
We define the standard echelon form of a matrix, SF(M), to be the echelon form in which each leading entry is positive and if a ij is a leading entry of the i th row, then 0 ≤ a rj ≤ a ij -1, 1 ≤ r <i. The standard echelon form of a matrix is unique. Note that the matrix in Eqn. (2) is not in standard echelon form, but the lower right hand corner 2 × 4 submatrix is in standard echelon form (see also Eqn. (3)).
Let M l (T) be the lower right hand corner 2 × 4 submatrix of M T in standard echelon form. If the endpoints arcs' unknowns, x1, x2, x3, x4 correspond to the four rightmost columns, then M l (T) is a tangle invariant. It is a tangle invariant in that if you take two diagrams of the same tangle T and place the endpoint arcs in the same order in the last columns of their respective coloring matrices, then no matter how the interior arcs are labeled, M l (T) will be the same for both diagrams. In addition, the absolute value of the determinant of the upper left 4 × 4 submatrix, d u (T) = 3, is also an invariant.
In the above example, the tangle diagram T is a 2-string tangle with six crossings. Hence its coloring matrix is a 6 × (6 + 2) = 6 × 8 matrix, and we were interested in the 2 × 4 matrix M l (T) as well as the determinant of the upper left 4 × 4 matrix. In the general case, suppose T is a diagram of an arbitrary n-string tangle with a k × (k + n) coloring matrix M T (listing the 2n endpoint arcs in the right-most columns of the matrix in a fixed order). Let M l (T) be the lower right-hand corner n × 2n submatrix of M T in standard echelon form, and let d u (T) be the absolute value of the determinant of the upper left (k - n) × (k - n) submatrix of M T . Both M l (T) and d u (T) are invariants of T . Note that columns corresponding to the endpoint arcs must be the right-most columns of the coloring matrix, and these columns must be in a fixed order when calculating M l (T). We will always order the endpoint arcs in a clockwise manner starting with a northwest endpoint arc.
In order to calculate M l (T) where T is an n-string tangle, we must label 2n endpoint arcs with distinct variables. If a string consists of just one arc (i.e., a string does not pass under any other string including itself so that it projects to just one arc; hence both endpoints of this arc lie on the boundary of the 3D ball), we can doubly label the arc, labeling one half of this endpoint arc x i , the other half x j , and adding the equation x i - x j = 0. Normally an n-string tangle with k crossings will have a k × (k + n) coloring matrix. But if any arcs are doubly labeled, then the coloring matrix will have more than k rows and (k + n) columns.
We describe a computational algorithm we have implemented to solve the system of tangle equations in Fig. 6. The full description is given in Methods. The majority of the algorithms were written so that this program can easily be modified to solve any system of n-string tangle equations up to around 8–10 crossings, including those modeling difference topology experiments applied to a protein complex that stably binds any number of segments of DNA.
# of Matrices Generated
Parity Fig. 10A
Parity Fig. 10B
total (≥ 8)
Recall that a tangle can be represented by a number of different diagrams related by Reidemeister moves. Unfortunately, there is no algorithm guaranteed to determine whether two tangle diagrams are equivalent. In fact, in order to simplify a diagram, it may be necessary to first increase the number of crossings in the diagram. Thus this software does not determine all tangle equivalences, but does reduce the output sufficiently to handle the remaining possibly equivalent tangles by hand. While generating matrices, we omit matrices where the corresponding diagram can be simplified by RI or RII moves (Fig.8). As discussed in Methods, we also perform some other simplifications which involve a combination of RI, RII, and RIII moves. As shown in table 1, this leaves us with 13 matrices that could correspond to tangles satisfying the system of equations in Fig. 6: ten with the parity shown in Fig. 10A and three with the parity shown in Fig. 10B.
We checked the remaining thirteen tangles corresponding to these matrices by hand. The ten tangles with Fig. 10A parity are all equivalent to the five crossing tangle found in  (Fig. 7A). The three tangles with Fig. 10B parity are all equivalent to one of the two eight crossing tangles in Fig. 7B, C. Recall that the two eight crossing solutions were not considered in  since the unknotted DNA substrate was negatively supercoiled and hence trapping left-handed crossings is biologically unlikely.
We have developed software to analyze the difference topology experiments in . Pathania et al  needed to assume the basic shape of a 3-branched supercoiled structure (Fig. 7) in order to find the solution shown in Fig. 1B (= Fig. 7A). With our software, no assumptions regarding the DNA conformation bound by the protein complex are needed except for an upper bound on the number of crossings. This algorithm can also be modified to analyze any difference topology experiment regardless of the number of DNA segments bound by the protein complex (although there is a bound on the topological complexity of the protein-bound DNA as discussed below).
A tangle solution is a topological approximation given as a 2-dimensional projection of a 3-dimensional structure. It does not determine sharpness of DNA bending, but it does give an important starting point from which other modeling techniques may be applied. Limited information regarding the Mu-DNA conformation existed before . Since then a partial structure based on scanning transmission electron microscopy (STEM) at cryo-temperatures has become available , but this involves only a portion of the protein complex and a change in the DNA sequences bound by Mu. Information regarding protein-bound DNA conformations can sometimes be obtained via crystallography, STEM, or FRET (fluorescence resonance energy transfer), but all these techniques are quite difficult and currently can only be applied to small protein-DNA complexes.
Recall that in the Mu tangle model from  (Figs. 5, 6), it is assumed that at most one crossing is trapped outside of the protein complexes (modeled within the green annulus). Since Mu and Cre bind to specific DNA sequences, the length of the DNA between the Mu binding sites and Cre binding sites can be controlled. The shortest length needed for the reaction to take place was determined in  in order to prevent trapping extraneous crossings. The difference topology experimental technique can also be applied to proteins that bind to arbitrary DNA sequences rather than specific DNA sequences, but the results would not be expected to be as clean (both in terms of experimental results as well as determining the appropriate tangle model). It was shown in  that if the length of DNA between binding sites is not properly controlled, then the number of protein-bound DNA crossings may be overestimated. But even if we are left with a topological approximation, it is still a significant improvement over having little or no information on how to draw the DNA in a protein-DNA complex.
We are not mathematically limited to equations resulting from Cre recombination. Any protein which can change DNA topology could potentially be used in a set of difference topology experiments to obtain a different system of tangle equations. For example topoisomerases change the topology of circular DNA by changing DNA crossings. It may be possible to obtain a more 3-dimensional model by averaging 2-dimensional projections of tangle solutions from two or more systems of tangle equations or tangle models [3, 4]. Cre, however, may be the easiest to work with due to its sequence specificity and its simple mechanism.
The software and its applicability to n-string tangle equations
Matrices which could correspond to coloring matrices of tangle diagrams are generated (see subsection Tangle generation in Methods)
The coloring invariants of each matrix are checked (subsection Checking the coloring invariants in Methods). Implementing this part of the software requires that we first mathematically simplify the system of tangle equations via 2-string tangle analysis (subsection 2-string tangle simplification in Methods).
Not all the matrices generated in step 1 will correspond to a tangle diagram. Hence each generated matrix is checked to determine if it actually corresponds to a tangle diagram (subsection Non-drawable matrices in Methods).
Different matrices can correspond to the same tangle. Thus we remove some (but not all) of the redundant matrices (subsection Equivalence moves in Methods)
No modifications are needed for Steps 1 and 3 in order to apply this algorithm to a different system of n-string tangle equations. For step 2, additional invariants may be needed in addition to or in replacement of the coloring invariants. Additional subroutines may also be needed for step 4.
Although coloring is not that powerful of a knot invariant, it is a powerful tangle invariant. As our results show, it is the only invariant we need to check to determine if a tangle up through eight crossings is a solution to the equations in Fig. 6. However, there is no guarantee that this invariant will be sufficient for a different system of tangle equations. Hence we may need to check additional invariants. Fortunately, there are a number of other invariants as well as software available for calculating these other invariants which can be used when needed [17, 21]. In particular we plan to add the homflypt polynomial knot invariant as an alternative option to the coloring invariant. The homflypt polynomial has been used in other algorithms requiring computational speed . Knots with nine or fewer crossings are uniquely identified by their homflypt polynomial. Hence if the knotted products of the difference topology experiments contain fewer than ten crossings, then checking the homflypt polynomial is sufficient (i.e., the homflypt polynomial will completely determine if a tangle is a solution to a system of n-string tangle equations if the equations only involve knots with less than ten crossings). Even if we need to use different invariant(s), this does not affect any other part of the algorithm. In particular, we can still use coloring matrices to encode tangle diagrams.
Our software left us with only 13 different coloring matrices which could correspond to tangle solutions to the system of equations in Fig. 6. We could have added additional methods to determine if two tangle diagrams are equivalent to further reduce this output, but it was quicker to check these 13 matrices by hand. For a different system of equations, additional methods to determine tangle equivalence may be needed to reduce the output to a handful of matrices. We will add additional subroutines to decrease the number of redundant tangles as needed.
The modifications that may be needed are straightforward. In fact they have been used by others for a computationally much more complex problem, knot tabulation . The techniques we use are very similar to those used to tabulate knots up through 16 crossings. The main difference between knot tabulation and our algorithm is that in tabulating knots, every knot diagram must be fully identified and all redundancies eliminated. In our algorithm, we discard diagrams that do not satisfy our equations, and hence only need to analyze a very small fraction of diagrams compared to the number of diagrams analyzed in knot tabulation. Also, since we focus on only a few systems of equations at a time, we can analyze by hand some redundancies among our tangle solutions. Hence we don't need to check nearly as many invariants or computationally determine as many tangle equivalences as in knot tabulation where millions of knots have been identified . Thus our algorithm is computationally much simpler than that required for knot tabulation.
Unfortunately, we cannot give a mathematical estimate regarding the number of solutions or the number of redundancies for an arbitrary system of tangle equations. In most cases, any modifications needed to reduce the number of repeated solutions will take at most a few days to implement. However, if the system of tangle equations is under-determined so that it has many small crossing solutions, then determining redundancies computationally will become much more important. An example of an under-determined system would be one modeling a partial set of difference topology experiments. In , Cre binding sites, in both inverted and direct orientations, were placed on each pair of the three loops emanating from the Mu transpososome. Hence six different substrates were constructed. If a protein binds, for example, four segments of DNA, then four loops will emanate from the protein-DNA complex. If Cre binding sites are placed on each pair of these four loops in both inverted and direct orientation, twelve substrates would be needed. In general if a protein-complex binds n segments of DNA, one would need to contract n(n - 1) different substrates if Cre binding sites are placed on each pair of loops in both orientations. An under-determined system would result if Cre binding sites are not placed on each pair of loops. We will eventually be able to solve under-determined systems for small crossing solutions as this problem is still much simpler than knot tabulation, but we expect this will take longer to implement.
Other mathematical methods
There are many mathematical techniques (for example [1, 24–28, 28–36]) as well as software [37, 38] for solving 2-string tangle equations. Hence many (but not all) biologically relevant 2-string tangle equations can be completely solved. Similar mathematics does not yet exist for solving n-string tangle equations for n > 2. Some work has been done on 3-string tangles  and solving 3-string tangles equations involving the class of 3-string tangles called 3-braids . There is also some work on classifying n-string tangles (for example, ). Also techniques in 3-manifold theory can be applied to solve n-string tangle equations for small crossing solutions , (Darcy IK, Luecke J, Vazquez M: A tangle analysis of the Mu transpososome protein complex which binds three DNA segments, manuscript in preparation). However, at the moment, there are no mathematical methods for solving the system of 3-string tangle equations in Fig. 6 or for most systems of n-string tangle equations.
Currently this C++ algorithm takes about two days on a Linux computer with AMD Opteron Processor (2.2 GHz cpu) to find solutions through eight crossings. However, the efficiency of the algorithm can be significantly improved by parallelizing it and running it on a cluster. Hence it should be possible to find solutions up to about ten crossings. As the number of tangles grows exponentially with crossing number, this algorithm can not be used to find high crossing solutions. Knots have only been tabulated up through sixteen crossings. Although our algorithm is computationally much simpler than knot tabulation, there are more tangles with k crossings than there are knots with k crossings. Hence we do not expect to be able to get much past ten crossings with a reasonable computation time.
Despite this computational limitation, we believe this algorithm is applicable to a wide array of protein-DNA complexes. The length of DNA bound by the protein limits the bound DNA's topological complexity. For example, the three DNA segments bound within the Mu transpososome are 50, 175 and 190 base pairs. However, we do not know of a theoretical tipper bound on the topological complexity of protein-bound DNA.
We believe eight crossings is a reasonable limit for the Mu transpososome. In addition to limits imposed by the lengths of the three protein-bound DNA sequences, the existence of a five crossing solution implies that a much more complicated solution with eight or more crossing is less likely. However, we have no proof that this is the case.
The computational algorithm described in this paper can be modified to solve any system of n-string tangle equations for small crossing tangle solutions. A long-term goal is to create software accessible to those without a background in knot theory. Eventually this software will be able to draw the tangle solutions. Some additional work is needed to handle under-determined systems of tangle equations as discussed above. But in the meantime if the system is not under-determined, we can readily modify this algorithm to solve any specified system of tangle equations up to around ten crossings; hence an experimentalist need not wait for the final version of this software before performing difference topology experiments.
Recall that the coloring equations (which correspond to rows in the coloring matrix) are determined by the crossings in the tangle diagram. Hence we determine the ordering of the rows by labeling the crossings. Beginning with string 1, we consecutively number the under-crossings (Fig. 12B). Hence for string 1, crossing number i occurs between string 1 arcs x i and xi+i. For string 2, crossing number j occurs between string 2 arcs xj+1and xj+2while for string 3, crossing number k occurs between string 3 arcs xk+2and xk+3. This determines the placement of the two 1's in each row (Fig. 12B). To generate matrices that could correspond to a coloring matrix, we can now place one -2 in each row in all possible combinations.
Not all matrices that could correspond to a 3-string tangle are generated (see below). Not all generated matrices correspond to a tangle (see section on Non-drawable matrices). Many different matrices correspond to the same tangle (see below and section on Equivalence moves).
Matrices not generated
The algorithm under discussion does not generate all matrices which could correspond to a tangle. A tangle diagram can contain an extraneous crossing manifested by the looping of a string over itself. If the loop does not pass under any string, this results in the equation x i - xi+1= 0. This is more general than an RI move (Fig. 8) as there could be strings passing under this loop. In any case this tangle diagram can be simplified, and hence we do not need to generate the matrix corresponding to this diagram. Since all matrices generated have two "l"s and one "-2" in each row, none of the matrices generated will correspond to a tangle containing such an extraneous crossing.
Another case that is not generated is the presence of a string not crossing under any arcs, and hence consisting of just one arc doubly labeled x i and xi+1. This case results in the equation, x i - xi+1= 0. We could easily generate this, but the system of tangle equations in Fig. 6 rules out such tangles as possible solutions.
The next part of this software checks the coloring invariant as this removes the majority of the matrices from consideration. However, for readability, we will discuss the drawability algorithm first.
In order to complete arc x3, we note that x1 crosses over between x3 and x4 (-2 appears in the third row and first column and hence the x1 is the overcrossing for this third crossing). Since x1 has already been drawn, we determine if the arc x1 is reachable from x3 by searching the region accessible to x3 (Fig. 14C, middle). In this case we see that x3 can reach x1 from both above and below and hence both cases are checked. Thus we draw the arc x3 approaching x1 from above in one case (Fig. 14C, top) and from below in the other case (Fig. 14C, bottom). We also draw the beginning part of the arc x4.
A portion of the arc x4 has been draw before (crossing over between x1 and x2), so we must determine if we can connect the previously drawn part of x4 with the beginning part of x4 that we just added. We determine if the previously drawn portion of x4 is within the region accessible to the newly drawn beginning part of x4 (Fig. 14D, left). Note that exactly one side of the previously drawn part of x4 is accessible. Hence there is exactly one way of connecting these two parts of x4 (Fig. 14D, right).
According to the matrix in Eqn. (4), the first string consists of exactly four arcs. Hence x4 must also connect to the boundary of the tangle ball. Therefore we check if the boundary of the tangle ball is accessible to the first part of x4 (Fig. 14E). It is not. After passing over between the arcs, x1 and x2, the arc x4 arc is trapped in the shaded region and cannot connect to the boundary of the 3-ball without introducing an extra crossing. Thus the matrix in Eqn. (4) does not correspond to a drawable tangle.
This is all done computationally. Currently no tangle diagrams are literally drawn. For a full description of the algorithm applied to link diagrams, see . The main difference between our algorithm and the algorithm in  is that since we are interested in tangles, we must consider the boundary of the tangle 3-ball as shown in the example in Fig. 14.
2-string tangle simplification
We can solve for the 2-string tangles in Fig. 15 using the tangle equations in Fig. 6. This step requires some mathematical background in tangle analysis, although there is software (available at KnotPlot.com) for solving some 2-string tangle equations . For information on how to solve 2-string tangle equations, see [1, 33]. For additional 2-string tangle analysis applied to the Mu transpososome, see (Darcy IK, Luecke J, Vazquez M: A tangle analysis of the Mu transpososome protein complex which binds three DNA segments, manuscript in preparation).
Checking the coloring invariants
Given a 3-string tangle T with k crossings, let M T be its k × (k + 3) coloring matrix. Let Op × (k-3)be a p × (k - 3) matrix with all zero entries. Suppose for some (k- 3) × (k-3) matrix A(k-3) × (k-3), 3 × 6 matrix M3 × 6 in standard echelon form and some (k-3) × 6 matrix B(k-3) × 6, SF(M T ) is as in Eqn. (5):
Connecting endpoint arcs x1 and x2 of T to obtain the 2-string tangle T 12 results in adding the equation x1 - x2= 0 to the matrix M T to obtain the matrix (Eqn. 6).
If T is a solution to the tangle equation in Fig. 20A, then this (k + 1) × (k + 3) matrix, , is a coloring matrix for T 12 . Since d u (T 12 ) = 1, we know that the upper left (k + 1 - 2) × (k + 3 - 4) submatrix of the (k + 1) × (k + 3) matrix SF() has determinant equal to 1. Since this matrix is in standard echelon form, this upper left (k - 1) × (k - 1) submatrix must be the identity matrix, I(k-1) × (k-1), which has 1's along the diagonal and zero's elsewhere. Thus A(k-3) × (k-3)is the (k - 3) × (k - 3) identity matrix, I(k-3) × (k-3). We also know that the lower right-hand corner 2 × 4 submatrix of SF() is equal to M l (T 12 ). Thus if T is a solution to the tangle equation in Fig. 20A, SF() is as in Eqn. 7 where * represents an arbitrary integer.
Hence, in order to determine if a matrix could correspond to a tangle, T, which is a solution to the tangle equation in Fig. 20A, we check if is row equivalent to one of the two matrices in Eqn. (7). This is not a guarantee that T is a solution as different tangles can have the same coloring invariants , but our computational results show that it is sufficient for solving the tangle equations in Fig. 18.
Alternatively, we can determine what the entries of the submatrix M3 × 6 of M T (Eqn. (5)) need to be in order for T to satisfy the tangle equations in Fig. 18. To determine M3 × 6, we add the equations x i - x(i+1) for each i = 1, 3, 5, and determine the constraints needed to satisfy the coloring invariants of Ti(i+1). If T satisfies the tangle equations in Fig. 18, then the determinant of A, the upper left (k - 3) × (k - 3) submatrix of M T , is 1 and M3 × 6 is as in Eqn. (8).
for some integer x, where r = 3 or -5, s = 2 or -4, and t = 2. As a check, both methods were implemented.
Recall that a tangle can be represented by a number of different diagrams related by Reidemeister moves. While generating matrices, we omit matrices where the corresponding diagram can be simplified by RI or RII moves and other matrix related moves (as described in the subsection Tangle generation). We also added two additional equivalence relations.
This also eliminates other tangle diagrams whose matrices contain these submatrices, but all such tangles can be simplified.
A tangle diagram containing the left-hand side of an RIII move (Fig. 8) will be equivalent to the tangle diagram obtained after the RIII move has been performed. Hence we choose one of these tangle diagrams and discard the other. After the above equivalence moves, we are left with thirteen possible tangles which can be checked by hand to determine if they correspond to equivalent or non-equivalent solutions to the tangle equations in Fig. 18 (or equivalently, Fig. 6).
This work was supported by a grant from the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology to ID and S. D. Levene. (NIH GM 67242) and by an Interdisciplinary Research Grant from The University of Iowa's Obermann Center for Advanced Studies to ID, S. D. Levene, and R. G. Scharein. We also thank John Luecke who ran a VIGRE REU at UT Austin which supported JC, ND, SM, and JS. ND and JS were also supported through the U.I. Undergraduate Scholar Assistant (USA) Program.
We also thank Abhijeet Bhutra for his contributions to the software discussed in this paper, Rob Scharein and KnotPlot.com for figures 1A, 7, 8, and Yosune Camio and Melanie DeVries for comments on an earlier version of this manuscript.
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