The unambiguous identification of links that are not topologically equivalent is of utmost relevance when studying links in a natural setting. Of special interest in the field of DNA topology is the action of enzymatic processes that produce DNA links. In this setting, one needs proper distinction between a link and its mirror image, or between two links related by reflection, orientation reversal, or component relabeling. For example, enzymes in the family of type II topoisomerases pass a segment of a DNA molecule through another thus introducing crossing changes (Figure 1
b). Another class of enzymes, site-specific recombinases, bind to two DNA segments, cleave and reconnect the ends (Figure 1
a). The local action of these enzymes on circular DNA molecules often results in global topological changes. Rigorous identification of the product links is used to study the topological mechanism of action of the enzymes. A mislabeling of the component orientation, or a mistaken chirality of the product can have severe implications on the mechanistic study. See more on this topic at the end of this section and in.
One of the primary goals of knot theory is to distinguish between link types. Knot theory is the mathematical study of links, i.e., embeddings of one or more disjoint circles in three-dimensional space. Each circle is a component of the link. A knot is a link with one component. Two links are topologically equivalent if there is an ambient isotopy between them. Intuitively, two links are equivalent if one can be smoothly deformed into the other without allowing phantom crossings between the curves. Each equivalence class is called a link type, and throughout this paper we may refer to an equivalence class as an isotopy class.
Traditionally, links have been tabulated following the Alexander–Briggs notation [1
] that organizes them by their minimal crossing number. The order of the links sharing the same crossing number is somewhat arbitrary. The link table in most common use is the Rolfsen table [2
]. There, links are labeled using Conway’s notation [3
] and an extension of the Alexander–Briggs notation [1
]. In the Rolfsen table, each knot diagram represents the unoriented knot K
and its mirror image
. For a link of two or more components, in addition to L
, the link diagram represents the link type with any component relabeling. However, the link L
and its mirror image
may be in different isotopy classes (i.e., may not be topologically equivalent). Such links are known as chiral links. Similarly, one could orient the link or label the components in several different ways which may yield non-equivalent links. Links which differ only in these ways share many properties, and are represented by a single unoriented link diagram in the standard link tables. In sum, the Rolfsen table provides no way to refer to chirality, orientations or component labeling [2
]. Hence, there has been a need to provide explicit diagrams in studies requiring any of this information. The objective here is to provide a link table that accounts for chirality, orientation, and component labeling.
One could of course compile yet another table with an arbitrary choice of mirroring, orientation, and component labeling motivated by an application at hand. Far from solving the problem, this would lead to further confusion. We instead propose to use geometric and topological properties of links to determine a standard diagram for each link. Our goal is to achieve a more clear relationship between isotopy classes of different link types. This is particularly useful when studying enzymatic actions and trying to establish relationships of link types before and after crossing changes or coherent band surgery (Figure 2
). More specifically, we propose to use writhe and linking number (introduced in Section 2
) to classify the different isotopy classes formed by mirroring, orientation reversal, and component relabeling.
Here by “classify”, rather than any deeper topological classification, we mean to distinguish isotopy classes of links in such a way that they can be consistently referenced. In Section 3
, we review previous efforts in this direction. In Section 4
, we propose a classification based on linking number and total writhe (defined in Section 2
) and define a canonical isotopy class for links. In Section 4
we define the BFACF algorithm. In Section 5
, we use numerical data obtained from BFACF simulations to estimate the total writhe for all prime links with up to nine crossings. In Section 6
we describe the numerical methods. We use self-avoiding polygons in
to represent knots and links with two components. For any given polygon length n
, we estimate the mean writhe values of all length n
representations of a given isotopy class. This extends previous work on knots [4
] and our numerical methods may be applied to any other link type.
Portillo et al. conjectured that given a chiral knot K
, the mean writhe of all length n
is bounded as n
varies, and furthermore is either positive for all n
or negative for all n
]. The numerical data in [4
] for knots with up to eight crossings supported this conjecture. Brasher et al. [5
] extended the numerical work to knots with up to 10 crossings; their data remained consistent with the conjecture. Table S6
is the table of canonical knot diagrams extended from that presented by Brasher et al. [5
]. We extend the conjecture to links as follows:
Given a c-component link L, the mean of the self-writhe for each component is bounded as the total link length n varies. Moreover, if the link is chiral, then the sum of these mean self-writhes is either positive for every n or negative for every n.
Here, the total link length n
is defined as the sum of the component lengths. Additionally, we could conjecture that if a link L
components has a property that exchanging component i
with component j
yields a different link (i.e., L
lacks exchange symmetry between components i
), then for any total link length n
, either the mean self-writhe of component i
is smaller than the mean self-writhe of component j
, or vice versa. We find, however, that, for smaller values of n
, this ordering may not be consistent. Specifically, minimum length conformations of the
link are provided as a counterexample (see Section 5
Based on Conjecture 1 and the supporting numerical data presented in Section 5
and in the supplementary materials
(online), we propose the following nomenclature. Let
be the set consisting of an oriented link and all links obtained from it by mirroring, component reversal and relabeling. We choose the standard link diagram L
representative of the set
as the one corresponding to an isotopy class where the sum of self-writhes and linking number are most positive, with components labeled in order of decreasing self-writhe.
We argue that this is a natural approach to choose the standard link diagram since writhe and linking number are very closely related to chirality (see Section 3.4
), and since we find that component self-writhe is related to exchange symmetries (i.e., component relabeling). Among the 2-component links with crossing number up to 9, only five are not fully disambiguated by our method. It is worth noting that previous approaches only partially disambiguate link isotopy classes (see Section 3.5
and Section 4.2
for more details). Additionally, the numerical methods used here to explore random conformations have been extensively used in studies of random knotting in the simple cubic lattice
], and in applications to DNA studies. For example to explore whether or not biological processes, such as those performed by topoisomerases and recombinases, are truly random or have some order [8
]. Relevance of proper link identification in DNA topology is discussed at the end of this section.
The structure of the paper is as follows. We start in Section 2
by defining writhe and linking number, which we will use to help distinguish the symmetry classes. Link symmetries and existing nomenclatures are reviewed and extended in Section 3
. We describe a systematic way to define a canonical isotopy class for each link (Figure A1
) in Section 4
. In Section 5
, we discuss the results of the numerical simulations used to distinguish between isotopy classes of links, and how they relate to Conjecture 1. Additional numerical results are included in the Supplementary Materials
online. In Section 5
, we also prove Theorem 2 that determines the difference in writhe between two polygons related by BFACF moves. This theorem deals with the boundedness of writhe for lattice links within the same isotopy class. The numerical methods used are described in Section 6
. The key outcome of our work is Figure A1
, the table of oriented link diagrams with labeled components based on our proposed nomenclature. For completeness, we have included Table S6
, the writhe-based knot table extended to 10 crossings based on the work of Portillo et al. and Brasher et al. [4
Importance of Link Symmetries in DNA Topology
Complete distinction between links related by reflection, orientation changes, and component relabeling is important in many problems in physics and biology. Our motivation for this study comes from the need to unambiguously identify knots and links arising from biological processes that change the topology of DNA. In its most common form, the B form, DNA forms a right-handed double helix consisting of two sugar phosphate backbones held together by hydrogen bonds. The backbones have an inherent antiparallel chemical orientation ( to ) and a circular molecule could be modeled as an orientable 2-component link where each backbone is represented by one link component.
More often, in DNA topology studies, the molecule is modeled as the curve drawn by the axis of the double helix. The axis can inherit the orientation of one of the backbones or be assigned an orientation based on its nucleotide sequence. In this way, one circular DNA molecule is modeled naturally as an oriented knot and two interlinked molecules are modeled as oriented 2-component links.
Different cellular processes can alter the topology of DNA. A notable example is that of replication of circular DNA. Replication is the process that makes two identical copies of a chromosome in preparation for cell division. If the chromosome is circular, as in the case of bacteria, replication gives rise to two interlinked chromosomes. If the original DNA circle is unknotted, then the newly replicated link is a right-handed torus link of type
]. The orientation given to the DNA circle before replication is naturally inherited by the components of the newly replicated link. Replication links are typically unlinked by enzymes in the family of type II topoisomerases which simplify the topology of their substrate DNA by a sequence of crossing changes. In [10
], Grainge et al. showed that in Escherichia coli
, in the absense of the topoisomerase Topo IV, replication links could be unlinked by site-specific recombination. Site-specific recombinases act by local reconnection, which can be modeled as a coherent band surgery on the substrate link (see Figure 1
a). This process was studied in [8
]. Importantly, the outcome of recombination can be dependent on the exact symmetry class of the link being acted on as illustrated in Figure 2
Furthermore, links arising as products of enzymatic reactions on circular substrates may have distinguishable components if the nucleotide sequence differs from one component to the other. In addition, some enzymes in the group of topoisomerases and site-specific recombinases have been found to have a chirality bias when identifying their targets (topological selectivity) or to tie knots or links of particular topology and symmetry type (topological specificity).