An algorithm based on 3-braids to solve tangle equations arising in the action of Gin DNA invertase

"The tangle model of Ernst and Sumners is an effective tool in the topological analysis of enzymes, a particular application of which aims at deducing the mechanism of action of site-specific recombination mediated by the Gin DNA invertase, an enzyme whose action involves 3-string tangles. In order to determine the enzyme’s mechanism of action, the tangle model yields equations that involve tangle indeterminates that must be solved for. While some of the available methods for solving such equations judiciously exploit the theory of 2-tangles, an algorithm is introduced in this note, based on 3-braid-theoretical results in [2], which allowed the authors to discover previously unreported solutions for the action of Gin DNA invertase. More generally, the algorithm allows one to solve 3-string tangle equations for 3-braid solutions under the assumption that each of the products of two or more rounds of recombinations is the unknot or a known 2-bridge knot different from the 2-component unlink. Rather than a specific language implementation, we here provide a pseudo-code description of the algorithm that permits its translation into a variety of computer languages and, possibly, its inclusion into more powerful analysis software.

Predicting Knot and Catenane Type of Products of Site-specific Recombination on Twist Knot Substrates

Site-specific recombination on supercoiled circular DNA molecules can yield a variety of knots and catenanes. Twist knots are some of the most common conformations of these products and they can act as substrates for further rounds of site-specific recombination. They are also one of the simplest families of knots and catenanes. Yet, our systematic understanding of their implication in DNA and important cellular processes like site-specific recombination is very limited. Here we present a topological model of site-specific recombination characterising all possible products of this reaction on twist knot substrates, extending previous work of Buck and Flapan. We illustrate how to use our model to examine previously uncharacterised experimental data. We also show how our model can help determine the sequence of products in multiple rounds of processive recombination and distinguish between products of processive and distributive recombination. This model studies generic site- specific recombination on arbitrary twist knot substrates, a subject for which there is limited global understanding. We also provide a systematic method of applying our model to a variety of different recombination systems.Comment: 17 pages, 13 figures. One figure correction. To appear in the Journal of Molecular Biology. See also arXiv:1007.2115v1 math.GT for topological proofs for the model presented her

Modeling knotted proteins with tangles

Although rare, an increasing number of proteins have been observed to contain entanglements in their native structures. To gain more insight into the significance of protein knotting, researchers have been investigating protein knot formation using both experimental and theoretical methods. Motivated by the hypothesized folding pathway of $\alpha$-haloacid dehalogenase (DehI) protein, Flapan, He, and Wong proposed a theory of how protein knots form, which includes existing folding pathways described by Taylor and B\"olinger et al. as special cases. In their topological descriptions, two loops in an unknotted open protein chain containing at most two twists each come close together, and one end of the protein eventually passes through the two loops. In this paper, we build on Flapan, He, and Wong's theory where we pay attention to the crossing signs of the threading process and assume that the unknotted protein chain may arrange itself into a more complicated configuration before threading occurs. We then apply tangle calculus, originally developed by Ernst and Sumners to analyze the action of specific proteins on DNA, to give all possible knots or knotoids that may be discovered in the future according to our model and give recipes for engineering specific knots in proteins from simpler pieces. We show why twists knots are the most likely knots to occur in proteins. We use chirality to show that the most likely knots to occur in proteins via Taylor's twisted hairpin model are the knots $+3_1$, $4_1$, and $-5_2$