Pathways of DNA unlinking: A story of stepwise simplification

Abstract

In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif-FtsK can remove replication links by local reconnection. We previously showed mathematically that there is a unique minimal pathway of unlinking replication links by reconnection while stepwise reducing the topological complexity. However, the possibility that reconnection preserves or increases topological complexity is biologically plausible. In this case, are there other unlinking pathways? Which is the most probable? We consider these questions in an analytical and numerical study of minimal unlinking pathways. We use a Markov Chain Monte Carlo algorithm with Multiple Markov Chain sampling to model local reconnection on 491 different substrate topologies, 166 knots and 325 links, and distinguish between pathways connecting a total of 881 different topologies. We conclude that the minimal pathway of unlinking replication links that was found under more stringent assumptions is the most probable. We also present exact results on unlinking a 6-crossing replication link. These results point to a general process of topology simplification by local reconnection, with applications going beyond DNA.

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Scientific RepoRts | 7: 12420 | DOI:10.1038/s41598-017-12172-2
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Pathways of DNA unlinking: A story
of stepwise simplication
Robert Stolz1, Masaaki Yoshida2,7, Reuben Brasher3, Michelle Flanner1, Kai Ishihara4, David J.
Sherratt5, Koya Shimokawa2 & Mariel Vazquez1,6
In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes
associated with replication and returning the newly replicated chromosomes to an unlinked
monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-
specic recombination complex XerCD- dif-FtsK can remove replication links by local reconnection.
We previously showed mathematically that there is a unique minimal pathway of unlinking replication
links by reconnection while stepwise reducing the topological complexity. However, the possibility
that reconnection preserves or increases topological complexity is biologically plausible. In this
case, are there other unlinking pathways? Which is the most probable? We consider these questions
in an analytical and numerical study of minimal unlinking pathways. We use a Markov Chain Monte
Carlo algorithm with Multiple Markov Chain sampling to model local reconnection on 491 dierent
substrate topologies, 166 knots and 325 links, and distinguish between pathways connecting a total
of 881 dierent topologies. We conclude that the minimal pathway of unlinking replication links that
was found under more stringent assumptions is the most probable. We also present exact results on
unlinking a 6-crossing replication link. These results point to a general process of topology simplication
by local reconnection, with applications going beyond DNA.
Flexible circular chains appear oen in nature, from microscopic DNA plasmids to macroscopic loops in solar
corona. Such chains entrap rich geometrical and topological complexity which can give insight into the processes
underlying their formation or modication. Knotted and interlinked states oen coincide with higher energy
states in physical systems and are usually undesired. Topology-simplifying reconnection processes involving
one or two cleavages are observed. Examples in biology include the action of type II topoisomerases and of
site-specic recombinases. Type II topoisomerases bind to two segments of double-stranded DNA, cleave one of
the segments, transport the other through the break (strand-passage) and reseal the break. Site-specic recombi-
nases bind to two specic sites (short segments of double-stranded DNA), introduce a double-stranded break on
each site, recombine the ends and reseal the breaks. e action of recombination enzymes is a local reconnection
event. We here investigate pathways of unlinking of newly replicated DNA links by local reconnection. e results
presented, and the numerical methods proposed are not restricted to the biological example and are applicable to
any local reconnection process.
In genetics, the observation of topological links dates back to studies in plants in the 1930s. In a study of chro-
mosomal variation in Crepis tectorum, M. Navashin observed ring chromosomes, noting “in one case, the two
daughter strands composing a normal chromosome failed to separate”. Navashin reported on a metaphase involv-
ing four rings, two of which were “united in the fashion of chain links”, thus documenting the appearance of two
newly replicated circular chromosomes forming a singly-linked catenane, or 2-crossing link1. In her study of ring
chromosomes in maize, Barbara McClintock observed the accumulation of several rings in the same cell and
hypothesized that “lack of uniformity in the splitting plane could give rise to a double sized ring with two inser-
tion regions or cause split halves of the ring to become interlocked”, thus introducing the ideas of chromosome
dimers and links (also called catenanes)2. ree decades later, DNA links were studied in vitro via random cycli-
zation of circular DNA in the presence of an excess of DNA circles3 and, in 1980 interlinked dimers formed by
nicked newly replicated 5.2 kb circular dsDNA mini chromosomes from SV40 were observed by electron
1Department of Microbiology and Molecular Genetics, University of California Davis, Davis, USA. 2Department of
Mathematics, Saitama University, Saitama, Japan. 3Microsoft, San Francisco, USA. 4Faculty of Education, Yamaguchi
University, Yamaguchi, Japan. 5Department of Biochemistry, University of Oxford, Oxford, UK. 6Department of
Mathematics, University of California Davis, Davis, USA. 7Present address: Takasaki City Oce, 35-1 Takamatsu-
cho, Takasaki, Japan. Correspondence and requests for materials should be addressed to M.V. (email: mrlvazquez@
ucdavis.edu)
Received: 28 June 2017
Accepted: 1 September 2017
Published: xx xx xxxx
OPEN
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microscopy4. e mechanisms of replication and segregation of circular DNA predict products that can be topo-
logically characterized as right-hand (RH) 2m-crossing torus links with parallel sites, which we here refer to as
parallel 2m-cats (denoted mathematically as parallel
m(2 )1
2
or
T m(2,2 )p
)5. ese topological forms were con-
rmed by characterizing the linked replication intermediates that accumulate in topoIV mutants6 (Fig.1(A)).
Sogo et al.7 hypothesized that catenanes appeared as replication intermediates of bacteriophage λ DNA
and observed that, in order to secure proper segregation of circular chromosomes at cell division, the linking
number of the two newly replicated molecules must be reduced to zero. However, the topology of a circular
double-stranded (ds)DNA molecule is insensitive to any manipulation that does not allow a double-stranded
break5. Nicking of a single DNA strand, however extensive, is insucient to unlink two newly replicated DNA
circles unless pre-existing nicks are present along the second strand. e type II topoisomerase topoIV is a
major decatenase in E. coli6,8. Grainge et al. showed that in the absence of topoIV, the XerCD- dif-FtsK molec-
ular machine can act in vivo to separate two interlinked, newly replicated chromosomes9. e XerCD complex
consists of the site-specic tyrosine recombinases XerC and XerD. e dif site is a 28 bp long recombination site
located within the terminus region of the E. coli chromosome. FtsK is a powerful translocase that assembles at
the division septum, where it activates XerCD- dif recombination. eir experimental data suggested a grad-
ual reduction in topological complexity of the substrates, which were RH 2m-cats with parallel dif sites9. e
proposed unlinking pathway, through which the enzymes unlink the replication links in a step-wise fashion is
illustrated in Fig.1A. In the gure, each closed curve represents a circular dsDNA molecule. e components of
a two-component link represent two newly replicated DNA chains.
Figure 1. (A) Under the assumption that each reconnection step strictly reduces the number of crossings of the
substrate, in Shimokawa et al.10 we showed that there is a unique unlinking pathway starting at a 2m-crossing
replication link. In E. coli a replication link is a 2m-cat with parallel dif sites6, and this pathway predicts the rst
product to be a
−m(2 1)1
knot with two dif sites in direct repeats. Two sites along a knotted chain are in direct
repeats if they induce the same orientation into the knot. Replication links are 2m-crossing right-handed torus
links with parallel sites (mathematical notation:
m(2 )1
2
). e pathway in the gure illustrates, for m = 6, the only
unlinking pathway starting at the parallel 2m-cat under the assumption that each reconnection step strictly
reduces the minimal crossing number. All the intermediate topologies are torus links
m(2 )1
2
or torus knots
−m(2 1)1
with two reconnection sites in direct repeats as in the gure. (B) One reconnection step: here the
cleavage regions of the reconnection sites on a
61
2
link are brought together to form a synapse (shown as a ball
enclosing two strings). e synapse is modeled mathematically as a 2-string tangle. In the case of XerCD site-
specic recombination, the strings in the tangle contain the core regions of the dif sites (indicated by two arrows
in a tangle P representing two very short segments of double-stranded DNA which physically behave as two
almost straight strings) and any bound DNA which does not change during recombination (gray shaded
region). Any interesting geometrical or topological complexity of the substrate is captured mathematically as an
outside tangle O that remains constant during reconnection. Before strand cleavage, the substrate is modeled by
the tangle equation
+=NO P()61
2
. e local reconnection is modeled by tangle surgery where P is replaced
with R, yielding a product represented as
+=NO RK()
, where K is a knot with two directly repeated sites. (C)
Local reconnection is a simple event which can be modeled as a band surgery, where P = (0) is replaced with a
tangle R = (w, 0) enclosing a vertical row of w twists, for some integer w. e rational tangle notation (or
Conway notation) for such vertical tangle is R = (w, 0). In the case when
=±w1
the notation simplies to
=±R(1)
. In the simplest cases, P = (0) with sites in parallel alignment goes to
=±R(1)
, and P = (0) with sites
in anti-parallel alignment goes to
=R(0,0)
as illustrated in the gure.
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A rigorous mathematical analysis of the recombination experiments of Grainge et al.9 showed that at least
2m steps are needed in order to unlink any RH 2m-cat with parallel sites10. is result relied simply on the
assumption that the XerCD tetramer binds the two dif sites and that a simple cut-reconnect-paste reaction ensues
(Fig.1C). If the shortest pathway of unlinking a 2m-crossing replication link has exactly 2m steps, it is natural
to ask how many such pathways exist and whether some are more likely than others. Under the assumption that
each step strictly reduces the topological complexity of its substrate (as measured by minimal crossing number),
Shimokawa et al.10 showed that the only possible pathway of unlinking a 2m-crossing replication link is that in
Fig.1A. Using tangle calculus, they proposed a 3-dimensional topological mechanism to take the parallel 2m-cat
to the unlink. is mechanism incorporates three solutions obtained by tangle calculus at each step of the process,
and the last three steps are fully characterized. e results in Shimokawa et al.10 provide unprecedented detail
in the study of the topological mechanism of DNA unlinking by site-specic recombination. Going beyond the
original problem of unlinking newly replicated circular chromosomes, these results apply to any reconnection
event that can be modeled using tangles as in Fig.1. For example, the same unlinking pathway proposed for DNA
links under site-specic recombination has been observed during reconnection events in physical elds such as
vortices in uid ow11–13. Further mathematical research on this subject can be found in the literature14–18.
Successful unlinking by XerCD-FtsK of newly replicated plasmids containing dif sites was shown in ref.9.
Quantication of these data gave weak justication to the assumption of stepwise reduction in complexity during
the unlinking reaction10. As can be seen in Fig.2, the gel quantication clearly illustrates the reduction of replica-
tion links by XerCD-FtsK site-specic recombination at dif sites. However, because of the complexity of the data,
in order to conrm stepwise reduction one would need to repeat the time course experiments9 for each individual
topology. is motivates the current work where we remove the assumption of stepwise decrease in complexity,
and design mathematical and numerical methods to assess the dierent unlinking pathways and the identication
of the most probable ones. We ask whether there are other minimal unlinking pathways and hypothesize that the
minimal pathway previously proposed9,10,19 and illustrated in Fig.1A is the most likely among all the possible
minimal pathways that arise. First, we allow the complexity of the products to decrease or remain the same at
each step of the reaction. We provide analytical proof that there are exactly nine minimal pathways of unlinking
a parallel 6-cat; many of the resulting transitions are fully characterized. Characterizing minimal pathways of
unlinking by local reconnection and resolving the topological mechanisms involved are problems of high theoret-
ical complexity since the number of possibilities quickly increases with the number of crossings of the substrate.
Likewise, characterizing the topological mechanism(s) taking a link Li to a knot Kj is equivalent to characterizing
all band surgeries between Li and Kj (see Fig.1C).
In order to discriminate between dierent minimal unlinking pathways for a given substrate and to extend the
study to higher crossing numbers, we eliminate the complexity assumption and develop a Monte Carlo method
to simulate local reconnection events. e method can be applied to a substrate with any topology, allows prod-
ucts of varying topological complexity, and facilitates the rigorous quantication of the transition probabilities
along each obtained pathway. Using this method we embark on a numerical study relevant to unlinking of DNA
replication links by site-specic recombination a dif sites. More specically, we restrict the numerical study to
knotted chains of xed length with two reconnection sites (representing the dif sites) that are evenly spaced along
Figure 2. Quantication of the time-course experiments9. e gel presented in Fig.1B in Grainge et al.9 showed
a time course of unlinking by XerCD-dif-FtsK50C at 25 °C of newly replicated plasmids containing dif sites.
Line scans of the gel were previously published10. In this gure each topological class is shown as a separate
series of points with linear interpolation. e caption assumes the bands observed correspond to the topologies
expected from a substrate composed of replication links, i.e. 2m-crossing links (e.g. 2m-cats), and some of the
corresponding knotted intermediates (open circle or 01, 31, 51). “Unlink” corresponds to the two unlinked
components in monomeric state (topology type
01
2
), and “Unknot” corresponds to the dimeric unknot (01). e
quantication clearly illustrates the reduction of replication links by XerCD-FtsK site-specic recombination at
dif sites. e complexity of the data is also evident, with the relative proportions of all the dierent topologies
uctuating from one step to the next, thus obscuring the signal.
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the chain, and linked chains consisting of the union of two circles of same length with one reconnection site in
each component. Details on the numerical experiments can be found in the Numerical Methods section and in
the Supplementary Methods.
e computational approach provides a rigorous means to discriminate between mathematically equivalent
unlinking pathways. e combination of the mathematical and computational studies provides strong quan-
titative support for the hypothesis that the unlinking pathway from Fig.1A is the most likely, even under the
weakened assumptions.
Nomenclature for knots and links
It is important at the outset to say a word about the naming convention used for the knots and links which arise
in this study (490 knots and 391 two-component links). A local reconnection event on a two component link with
one cleavage site in each component yields a knotted chain with two sites in direct repeats (cf. Fig.1A). Rolfsen’s
Knot Table20 summarizes the knot nomenclature used in the mathematics community, which was not intended
to distinguish between mirror images nor between oriented links, an important consideration when dealing with
circular DNA and other biopolymers. Chirality is relevant, and indeed crucial, to characterize biological and
chemical compounds. In this paper, we use the writhe-based knot nomenclature proposed in Brasher et al.21. e
writhe is a geometrical invariant that provides a measure of a chain’s entanglement complexity and chirality. It is
computed analytically using a Gauss double integral and can be estimated numerically by taking the average of
the writhe of a planar diagram taken over all projection directions (the projected writhe). e mean writhe of a
knot K refers to the average of the writhes of all knotted chains of type K. Numerically this is estimated by aver-
aging over a suciently large, randomly generated ensemble of conformations of type K. A representative of a
chiral pair is chosen based on its mean writhe21. We extend this nomenclature to the 2-component links depicted
in Fig.3. For prime 2-component links with 9 or more crossings we use the default notation from Knotplot22. For
more details and a comparison with other published nomenclature for links refer to the Supplementary Methods
and to Supplementary Fig.S5.
Results
There are exactly 9 shortest pathways to unlink the 6-cat that do not increase substrate com-
plexity. We consider an event where two oriented sites come together and undergo cleavage followed by
reconnection. If the substrate is a single circle, then the oriented sites are in direct repeat, i.e. they induce the same
orientation into the circle. If the substrate consists of two circular chains, then there is one site in each chain. Note
Figure 3. (A) Illustration of some of the knots relevant to the present study and their nomenclature. e
chirality is consistent with that in Brasher et al.21. e green arrows along the unknot 01 represent the two
reconnection sites. e sites shown are equidistant and in direct repeats. A complete table of prime knots with
up to 10 crossings and information on how they compare to those in Rolfsen20 can be obtained from the authors
upon request. (B) Nomenclature for two component links relevant to the present study. e green arrows
represent the reconnection sites, which confer an orientation to each link component. e nomenclature is
described in the Supplementary Methods and in Supplementary Fig.S5. For 2-component links with 9 or more
crossings we revert to the default Knotplot naming convention. (C) e four possible combinations of chirality
and orientation for the 4-crossing torus link. A comparison between the nomenclature used in this paper and
that in Rolfsen20 and in works by Kanenobu28,29 is included in Supplementary Fig.S5. Arrows indicate the
relative orientations of the sites.
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that such an event always changes the topology of the substrate: reconnection between two sites in separate com-
ponents of a link yields a knot with two sites in direct repeats, and reconnection on a knot with two directly
repeated sites yields a 2-component link with one site in each component. e reconnection event is modeled as
a system of tangle equations as described in Fig.1(B). In the context of DNA unlinking, as in Shimokawa et al.10,
we model dsDNA as a curve dened by the axis of the DNA double helix, and the synapse formed by the enzymes
bound to the core regions of the dif recombination sites as the 2-string tangle P. Reconnection changes P into R.
If we assume that each reconnection is modeled as a coherent band surgery, i.e. P = (0) and R = (w, 0) for some
integer w, then any minimal pathway to unlink an n-crossing torus link with parallel sites (e.g.
41
2
or
61
2
) has
exactly n steps. Furthermore, if each reconnection step is assumed to strictly reduce the complexity of its sub-
strate, then the minimal pathway is unique: i.e. RH 2m-cat, RH
−m(2,2 1)
-torus knot, RH
−m(2 2)
-cat,

, RH
trefoil, Hopf link, trivial knot, trivial link. Figure1A illustrates the 6-cat case. Since the experimental data9 only
gives weak support to the assumption that the complexity goes strictly down at each step of the reaction (Fig.2),
we here examine the case where no reconnection step increases the number of crossings and provide analytical
characterization of all shortest pathways from the 6-cat to the unlink.
Assumption 1. Consider a reconnection pathway from a parallel RH 2mcat to the unlink. Assume that each prod-
uct along the pathway is a knot or a 2-component link, that the pathway is shortest, and that no reconnection event
increases the number of crossings of its substrate.
Recall that any shortest reconnection pathway from
m(2 )1
2
to the unlink has exactly 2m steps10. In eorem 2
we show that there are exactly nine unlinking pathways satisfying Assumption 1.
eorem 2. A pathway from the parallel RH 6-cat that satises Assumption 1 is one of the 9 shown in Fig.4.
e 9 pathways found in eorem 2 involve 16 possible transitions taking a knot to a link or vice versa; 6 of the
transitions have fully characterized mechanisms. e proof of the theorem and the characterization of the mech-
anisms are presented in the Supplementary Methods. Figure4 summarizes the results as an oriented graph where
each node is a knot/link type and each edge represents the transition between two topologies by one reconnection
step. All minimal pathways taking the parallel
61
2
to the unlink
01
2
, and satisfying Assumption 1 are shown. In the
next section we undertake a thorough computational study with the objective of discriminating between minimal
pathways while minimizing the number of assumptions. In particular, we use the numerical work to assign fre-
quencies to each transition in the pathway graph (represented in Fig.4 as weights on the edges).
We here give a dra of the proof of eorem 2. More details, including Lemmas S1-S8, Propositions S9-S17,
and FigsS1 and S2 exhibiting the steps of the proof and relevant band surgeries for each of the transitions in
Fig.4, are included in the Supplementary Methods. In order to characterize the minimal pathways starting from
Figure 4. e substrate at the top le corner is the link
61
2
with two reconnection sites in parallel orientation.
e pathways are represented as an oriented graph where the nodes are the knot or link types, and two nodes
are connected by an edge if one can go from one to the other via a reconnection event. e substrate and
product of each reconnection are indicated by the orientation of the edges. e diagrams above each edge
illustrate an example of the corresponding reconnection event by showing the band where the band surgery will
be performed. e weights on the edges correspond to transition probabilities obtained numerically. Details of
the simulations are in the Numerical Methods section below, and in the Supplementary Methods and
Supplementary Data.
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the parallel
61
2
link, we rst investigate the eect of band surgeries on certain topological invariants such as the
signature, the Jones polynomial, the Q polynomial and the Arf invariant of the knots and links involved in those
pathways. By Lemma S6, the sequence of the signatures of knots and links is −5, −4, −3, −2, −1, 0, 0. Lemma S7
shows that split links can not appear in a shortest pathways. Lemma S8 identies the candidate topologies for the
minimal pathways from
61
2
.
Outline of the proof
(First step) From Proposition S9, the product knot obtained from
61
2
is either 51 or
3#3
11
.
(Second step) From Proposition S10, the product link obtained from 51 is either
41
2
or
3#2
11
2
. From Proposition
S11, the product link obtained from
3#3
11
is either
63
2
or
3#2
11
2
.
(ird step) From Proposition S12, the product knot obtained from
63
2
is 52. From Proposition S13, the prod-
uct knot obtained from
3#2
11
2
is either 52 or 31. From Proposition S14, the product knot obtained from
41
2
is 31.
(Fourth step) From Proposition S15, the product link obtained from 52 is either
21
2
or
′
⁎
41
2
. From Proposition
S16, the product link obtained from 31 is
21
2
.
(Fih step) From Proposition S17, the product knot obtained from
′
⁎
41
2
is 01. e product obtained from
21
2
is
01. In the last step, the recombination event changes 01 into
01
2
. ese steps cover all transitions satisfying the
Assumption 1.
Topological mechanisms of reconnection. e topological mechanisms of events between the following
(substrate, product) pairs have been fully characterized10:
(3 ,2),(2 ,0),(0 ,0)
11
21
2111
2
. e topological mechanisms
between pairs
′
⁎
(5 ,2),(5 ,4 )
21
221
2
,
′
⁎
(4 ,0)
1
21
are characterized in the proposition below. For all transitions along
the 9 minimal pathways, Fig.4 illustrates one possible band surgery relating the knot to the link. e proof of
Proposition 3 is given in the Supplementary Methods, Characterization of Mechanisms section (Supplementary
Fig.S3, Proposition S18, eorem S19, Lemma S20).
Proposition 3 A23. Suppose
+=NO P()52
,
+=NO R()21
2
, P = (0) and R = (w, 0). en =−−
()
Ow
7
72
.
B23. Suppose
+=NO P()52
,
+=′
⁎
NO R()41
2
, P = (0) and R = (w, 0). en =−−
()
Ow
7
74
.
C24. Suppose +=
′
⁎
NO P()41
2
,
+=NO R()01
, P = (0) and R = (w, 0). en =
−−
()
Ow
4
41
.
Because XerC and XerD are tyrosine recombinases and act through a Holliday Junction Intermediate, the
tangle pairs (P, R) that are relevant to unlinking of DNA replication links by Xer recombination are
=−PR(, )((0),(1))
p
,
=PR(, )((0),(0,0))
a
=PR(, )((0),(1))
p
as illustrated in Fig.1C. e above proposition
allows to determine all the topological mechanisms for each of the three combinations of substrate and product
in the statement. We illustrate the solutions in Proposition S18 and in Supplementary Fig.S3 in the Supplementary
Methods. Just as in Shimokawa et al.10, here each system of tangle equations yields three solutions, and the three
solutions can be interpreted as representing a unique 3-dimensional topological mechanism.
Which unlinking pathways are most probable? In the previous section, we proved analytically that
under Assumption 1 there are 9 minimal pathways of unlinking the parallel 6-cat,
61
2
. e mathematical analysis
that includes enumeration of pathways and characterization of topological mechanisms becomes dicult for
substrates with high crossing numbers. Furthermore, if the assumption of reduction in complexity–which is
equivalent to imposing a topological lter in the physical system–is lied, then the number of possible pathways
increases rapidly and the detailed mathematical analysis quickly becomes intractable. We here remove
Assumption 1 and set out on a numerical exploration of reconnection pathways starting from a broader set of
substrate topologies. We develop soware which nds reconnection sites along polygonal chains in the simple
cubic lattice and simulates the reconnection event. Figure5C illustrates the basic reconnection move on a simpli-
ed polygon. Figure5A shows a lattice trefoil with one single reconnection site, before and aer local reconnec-
tion. We simulate reconnection to explore dierent topological transitions, to quantify transition probabilities
and to discriminate between unlinking pathways that are mathematically indistinguishable when only substrate,
product and length are specied.
We provide numerical evidence that, of all minimal pathways starting with the RH parallel 6-cat, the one in
Fig.1A is the most likely. e weights in Fig.4 correspond to the transition probabilities obtained in the numeri-
cal simulations. More generally, our numerical data suggest that this trend holds for any substrate that is a RH
2m-cat with parallel sites, or a RH
−m(2 1)
-torus knot with two sites in direct repeats. It is important to empha-
size that the simulations do not use Assumption 1. Figure5B is a circos gure that shows all observed reconnec-
tion transitions that maintain or decrease minimal crossing number and that belong to an observed minimal
pathway from the 91 knot. e thickness of the arcs corresponds to the directed transition probability between
two topologies. Transitions in the most probable minimal pathway from 91 are colored red. e predominance of
these most probable unlinking pathways is consistent with the experimental observations for XerCD-FtsK- dif
site-specic recombination on DNA replication links9, and for reconnection in uid vortices12, and is also consist-
ent with the predictions in the literature10,11.
e minimum distance between the link type Li and the knot type Kj in terms of band surgeries is called nulli-
cation distance25,26. In the numerical experiment we started by choosing knots and 2-component links that are at
nullication distance 1–3 from one of the 11 knots or links along one of the 9 minimal pathways of eorem 2 and
Fig.4, or are obtained from these topologies by taking mirror images or reversing the orientation of one of the
components. For completeness, we expanded the initial set to include 491 substrate topologies representing
almost all knots and links with 9 or fewer crossings. Reasons for omitting a handful of 9-crossing split links from
the substrate set are described in detail below. We use the BFACF algorithm to generate large independent
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ensembles of conformations for each substrate topology. BFACF is a dynamic Monte Carlo method which sam-
ples uniformly the set of all lattice polygons of xed topology for a given mean length27. e BFACF moves used
to perturb each chain are illustrated in Fig.S4 in the Supplementary Methods. Split links such as the unlink
01
2
or
∪
03
11
(see Fig.3), even though they appear as reconnection products, are not used as substrates due to the dif-
culty of keeping the components together without altering the Monte Carlo procedure. In order to improve the
eciency of sampling statistically independent conformations we implemented BFACF as a Composite Markov
Chain (CMC). Details of the simulations, including a description of the algorithms and dierent parameters, are
included in the numerical methods section and in the Supplementary Methods. Fig.S6 in the Supplementary
Methods illustrates all the transitions observed between 881 topologies in the numerical experiment, including
those that do not appear in minimal pathways from 91. e resulting transition probabilities are available in
matrix form in the data spreadsheet provided as Supplementary Information (Supplementary Data).
Figure5D contains exact counts for the number of minimal unlinking pathways for torus knots and links with
up to 6 crossings, and the corresponding numerical estimates for 7 and 8 crossings. Under Assumption 1 there are
9 minimal pathways of unlinking the
61
2
link. In the numerical study, we nd 36 minimal unlinking pathways for
the 71 knot and 208 minimal unlinking pathways for the
81
2
link, under Assumption 1 (
PL()
min
). Once the
Assumption is removed, we observe
=P(7)2760
1
minimal pathways for the knot 71 and
=P(8 )6434
1
2
minimal
pathways for the link
81
2
(in this case the crossing number can increase at any given step). However it has been
shown analytically that there are innitely many possible minimal pathways between any 2n torus link with par-
allel sites and the unlink17. e numerical data can provide biologically-relevant information by establishing a
ranking of the most likely pathways. e third row in Fig.5D indicates the number of distinct product topologies
(as detected by the HOMFLY-PT polynomial) observed for torus knots and links of the type
T n(2,)
with 8 or
fewer crossings aer a single reconnection step.
Figure 5. (A) e substrate (le) is a lattice trefoil with 120 segments and two directly repeated reconnection
sites indicated by a white sphere. e product (right) is a 2-component link obtained aer one reconnection
event. All substrate knots have directly repeated sites that are 60 segments apart, with a tolerance of ±6
segments, and all links have two components 60 ± 6 long so that the sum of the lengths is exactly 120.
Reconnection on links is only performed between sites in dierent components. (B) Circos gure: all
reconnection transitions in a minimal pathway from the 91 that satisfy Assumption1. 2-component links (resp.
knots) are arranged by increasing crossing number from bottom to top in the le (resp. right) hemisphere, and
are color-coded blue (resp. red). Color intensity increases with decreasing crossing number. An arc between K
and L indicates at least one observed reconnection event between K and L. e thickness of the arcs corresponds
to the directed transition probability between two topologies. Transitions with an observed probability <0.2 are
thickened to be more visible. Transitions are colored according to the probability of the most probable minimal
pathway they are a member of. e rst, second, and third most probable unlinking pathways from 91 are
colored red, orange, and yellow, respectively. If no arc appears between a pair {K, L}, this means that no
reconnection between them was observed. Observed transitions for all substrate topologies, including those in
non-minimal pathways, are included in Supplementary Data and in Fig.S6 in the Supplementary Methods. (C)
Local reconnection move between two directly repeated sites. In the juxtaposition the reconnection sites,
indicated with hashed lines, are at distance 1 and in antiparallel alignment. (D) L are T(2, n) torus knots and
links (Fig.1).
PL()
min
is the number of minimal unlinking pathways observed for L under Assumption1. P(L)
indicates the total number of minimal pathways observed for L without Assumption1. It is known that there are
innitely many minimal unlinking pathways for any
T n(2,2 )
link with parallel sites17.
−
NHOMFLY PT
is the
number of distinct HOMFLY-PT polynomials observed aer one reconnection.
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Discussion
In eorem 2 we prove that there are exactly 9 shortest unlinking pathways for the
61
2
, assuming that at every step
the complexity of the substrate goes down or remains the same. e 9 pathways are illustrated in Fig.4. We solve
the topological mechanisms involved for 6 of the 16 steps along these pathways. We develop a new Monte Carlo
based numerical method which allows us to model local reconnection on chains of xed length and topology. We
run the numerical simulation on each topology found to be within 3 nullication steps from any topology in
Fig.4. Notice that in these experiments there is nothing preventing the complexity of a substrate from going up at
any given step. We can determine the set of all minimal pathways from any of the substrate topologies, and single
out the most probable pathway. In Fig.5 we provide numerical estimates for the number of minimal pathways for
torus knots and links with 7 and 8 crossings. In our numerical data the most probable minimal pathway from a
torus link (or knot) to the unlink is the one where every intermediate is also in the torus family as in Fig.1A. e
data from the numerical experiments can be found in the Supplementary Data.
Mathematically, extending eorem 2 to determine all minimal pathways for T(2, N) torus knots and links is
dicult. In general, if the substrate is a torus knot or link T(2, N) one can nd multiple pathways that preserve the
minimal crossing number at many steps. e complexity of the problem grows with the minimal crossing number
of the substrate. For example, using numerical simulation we estimate the number of minimal pathways from the
71 (resp.
81
2
) to the unlink to be at least 36 (resp. 208) under Assumption 1. ese are not tight bounds due to the
limitations with using links of the form
K#21
2
as substrates in the numerical experiments. It is known that when
the assumption is removed, there are innitely many shortest pathways between the
T N(2,2 )p
torus link and the
unlink17. In our numerical work, once Assumption 1 is removed we count at least 744, 2760 and 6434 shortest
unlinking pathways for
61
2
, 71 and
81
2
, respectively.
e problem of computing the nullication distance between a knot and a link is of interest to the mathemat-
ical community17,25,26,28,29. In cases where the analytical tools fail to provide an exact nullication distance, one
can estimate the distance between two topologies using the numerical method and possibly remove ambiguities
by exhibiting the relevant band surgeries.
e numerical simulations in this study posed a number of challenges. For example, in order to generate an
ensemble of essentially independent unknots 01 of length 120 we had to go through at least twice as many itera-
tions of the BFACF algorithm than for any other substrate topology. Further, these unknots contained synapses
meeting the reconnection criteria approximately once every 7.5 × 109 iterations. In order to improve the eciency
of such runs, we implemented the BFACF algorithm as a Composite Markov Chain process30–33. Similar chal-
lenges extend to any topology consisting of a connected sum of a knot and a Hopf link
K#21
2
, or the disjoint union
of a knot and an unknot
∪
K01
(see examples in Fig.3). In the rst case, the unknotted component tends to
shrink, making it dicult to satisfy the equal-length criteria for recombination. In the second case, even though
these topologies appear as reconnection products, they cannot be used as substrates due to the diculty of keep-
ing the components together (without biasing the simulations for those specic substrates). Now consider an
example where a bacterial chromosome dimer forms a 31 knot with two equidistant directly repeated dif sites. In
our simulations we see that 0.025% of trefoils transition to
∪
03
11
, the disjoint union of an unknot and a trefoil,
and 95.2% of trefoils transition to
21
2
. In the rst case the knotted dimer is eectively unlinked in one step, but one
of the components will remain knotted, which can pose problems during chromosome segregation. In the second
case unlinking of the trefoil can be achieved in 3 steps, with a combined probability of 0.925; the nal product is
01
2
, a union of two circles which can then segregate at cell division.
In the case of unlinking of DNA replication links, each component of the link corresponds to a newly repli-
cated chromosome from E.coli with one dif site in each component. is example motivated our choice to let two
reconnection sites within a single circle be equidistant, and the two components of a linked product or substrate
have the same length. In dierent contexts, such as that of site-specic recombination between non-equidistant
sites, more general homologous recombination, and possibly other reconnections in physics, the distance between
sites will be an important parameter, requiring further exploration of the length and topology dependence of the
transition probabilities obtained by the numerical method.
Furthermore, in nature, DNA molecules are oen found tightly packaged in crowded environments. A study
of reconnection on conned chains would shed light on whether connement plays a role in driving topologi-
cal simplication by any process involving local reconnection. Existing studies of the connement of polygonal
chains inside and outside the lattice suggest methods for generating ensembles of conformations34,35.
Materials and Methods
Mathematical Methods. e tangle method is briey summarized in Fig.1. e naming convention used
for knots and links is reviewed in the Introductionand in Fig. 3. More detailed mathematical methods and results
used in the proof of eorem 2 are provided in Fig.4 and in the Supplementary Methods. A site-specic recom-
bination event is modeled as a local reconnection and is represented mathematically as a system of tangle equa-
tions as described in Fig.1B. e circular chain represents the starting knot or link, and P is a 2-string tangle that
encloses the reconnection sites. Reconnection changes P into R. We assume that each reconnection is modeled as
a coherent band surgery, i.e. P = (0) and R = (w; 0) for some integer w (Fig.1C).
Numerical Methods: modeling reconnection. Computer simulations of local reconnection. We use an
integrated set of computational tools to generate and lter ensembles of conformations, perform reconnection,
identify product topologies, generate transition probabilities and facilitate statistical analysis of the results. Given
an ensemble of lattice conformations with xed length and constant topology, our algorithm searches for possible
synapses along each conformation, selects one uniformly at random, and performs reconnection as illustrated in
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Fig.5A. Our original motivation came from XerC/D site-specic recombination at dif sites in newly replicated
chromosomes with one site in each component or in chromosome dimers with two equidistant directly-repeated
sites. In this case reconnection events are constrained by the position and orientation of the dif sites. We therefore
impose a set of constraints on where to perform reconnection. ese can be seen as topological lters that can
be adjusted to best t the scenario to be modeled. Here, a reconnection synapse is dened as a pair of coplanar
edges of distance one apart with antiparallel orientation; each of the two oriented edges is a reconnection site.
Reconnection exchanges each edge of the synapse for one perpendicular to it as shown in Fig.5C. e set of pos-
sible edge pairs on which to form a synapse is further constrained by step distance along the conformation. Here
we adjust this parameter to constrain the location of the synapse so that the arc lengths on each side are equal
within a ±6 range, while enforcing the total length of the knotted polygon, or the sum of the lengths of the com-
ponents of interlinked polygons, to be xed. For knots this models two equidistant sites in the synapse. For two
component links, it models two components of equal length with a single site in each of the two components. We
exclusively sampled conformations of total length 120 which contain at least one reconnection synapse.
Generation of reconnection substrates. Self-avoidance is an important property when modeling biopolymers
such as circular DNA. Here, conformations in the simple cubic lattice,
Z3
, are self-avoiding polygons whose verti-
ces have integer coordinates and whose edges are parallel to one of the three coordinate axes. e BFACF algo-
rithm is a dynamic Monte Carlo method which samples from the space of lattice conformations of a fixed
topology27. e states of the resulting Markov Chain are conformations obtained by rst randomly selecting an
edge, then attempting one of the three moves shown in Fig.S4 in the Supplementary Methods ((−2)-move,
(+2)-move or (0)-move). None of these moves can ever change the link type of the conformation27,36.
Generating large ensembles of conformations for each topology with at least one valid synapse posed signi-
cant technical challenges. e 01 knot and links of the type
K#21
2
where K is a knot with high crossing number
were particularly problematic. is is because the component with trivial topology tends to have a short average
length, making sampled conformations that form a reconnection synapse very rare. For example, the 01 forms
such a synapse in fewer than 1 in 1.3 × 106 sampled conformations. To address these challenges and gain the
computational performance needed for this study, we here extend the ecient, constant time (in knot length)
implementation of the BFACF algorithm used in previous work34,35,37,38 by employing it as a Composite Markov
Chain (CMC) Monte Carlo process30–33,39. CMC BFACF iterates simultaneously on multiple Markov chains with
dierent fugacity parameters, swapping conformations between chains when certain weighted random criteria
are met; more details of the implementation are included in the Supplementary Methods. CMC Monte Carlo
improves eciency by exchanging conformational states between chains, thus improving the speed at which the
conformations are randomized. We sample conformations at a frequent xed rate and correct for dependent
samples using block mean analysis40, therefore standardizing the sampling methodology across all of the topolo-
gies in the study and avoiding reliance on direct estimations of integrated autocorrelation time. With this meth-
odology, we generated in the range of 107 conformations for every substrate topology. Of the topologies for which
a reconnection event was observed, the number of conformations containing at least one reconnection synapse
ranged from approximately 1.5 × 106 for the
⁎
913
knot, to as little as 86 for the
6#2
21
2
link. Two component topolo-
gies in which the two components are of dierent topology are dicult to sample eciently because of the rarity
of conformations that meet our stringent arclength criteria. Split links, i.e. those topologies in which the two
components are not interlinked, are even more problematic because both components tend to travel away from
each other, thus dramatically reducing the probability of sampling conformations that contain a valid synapse. We
identied those topologies as products of reconnection, but did not include them in the set of substrate topologies
described in the next paragraph.
Recall that 9 minimal unlinking pathways from the 6-cat were obtained analytically in eorem 2 under the
assumption that each reconnection step either preserves or reduces the complexity of the substrate. Our simula-
tions eliminate that assumption, enabling wider exploration of possible topological reconnection pathways. We
start with 491 substrate topologies, including those along the 9 unique pathways from Fig.4 (excluding the unlink
01
2
). With CMC BFACF we generate ensembles of conformations with xed topology to be used as reconnection
substrates. e number of substrate conformations generated ranges from 1.2×107 for the
76
2
link, to more than
6.9 × 108 for the 01. We perform one reconnection per conformation and identify the resulting topology. Including
all substrate topologies and the identiable products aer reconnection, there are 881 topologies being analyzed
in the study (490 knots and 391 two component links).
Knot identication. Our simulations require a rigorous, unambiguous way of identifying the knot or link con-
formation types in
Z3
. With the exception of chiral knots 817 and 942 which have the same HOMFLY-PT as their
mirror images, and 912 which has the same HOMFLY-PT as 41#52, all prime knots with nine or fewer crossings
can be unambiguously identied using the HOMFLY-PT polynomial41,42. Our knot identication soware is
based on the other published algorithms43,44. In order to identify product topologies, we rst perform 20,000
BFACF iterations with randomly chosen (0) and (−2) moves. At each step, the conformation either remains the
same length or becomes shorter, in many cases approaching the minimal length for that topology38. e nal
conformation goes through an energy minimization algorithm22, we compute an extended Gauss code and iden-
tify the topology using the HOMFLY-PT polynomial. Information on those oriented knots or links with 10 or
fewer crossings that HOMFLY-PT fails to identify uniquely is included in the Supplementary Methods.
Recombination between two directly repeated sites along a single circular chain yields a 2-component link.
e number of product topologies increases dramatically with the complexity of the substrate. Figure3 shows
a selection of some of the expected products, including composite links that are not normally shown in knot
tables. Composites are of two types: connected sums of prime knots or links; and disjoint unions. In this study,
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we perform recombination on two types of substrates: (i) knots with two (approximately) equidistant directly
repeated sites; and (ii) links with 2 components of identical total length and with one site in each component.
More specically, each substrate knot is a self-avoiding lattice polygon of length 120 and recombination occurs
on two directly repeated sites that are between 54 and 66 units apart (Fig.5A). Each linked substrate consists
of two self-avoiding polygons between 54 and 66 units long, such that the sum of their lengths is exactly 120.
Recombination is restricted to synapses where two sites, one in each component, are found at unit distance apart
and in anti-parallel alignment as illustrated in Fig.5(A and C). A small representative subset of the knot and link
types used in the simulations is shown in Fig.3, and the naming convention is described in the nomenclature
section, in the Supplementary Methods and in Supplementary Fig.S5.
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Acknowledgements
is research was supported by the following: Japan Society for the Promotion of Science KAKENHI grant
numbers 25400080, 26310206, 16H03928, 16K13751, 17H06463(to K.S.), 26800081 (to K.I.); National
Science Foundation DMS1716987 (MF,MV) and CAREER Grant DMS1057284 (MV, RS, MF, RB) and NIH-
R01GM109457 (MV); Welcome Trust SIA 099204/Z/12Z and 200782/Z/16/Z (DJS). e authors are grateful to R.
Scharein for providing assistance with Knotplot and for his work on the rst version of the reconnection soware;
C. Soteros, M. Szafron and M. Schmirler for contributing their statistical expertise; J. Arsuaga, D.W. Sumners and
S. Witte for helpful discussions; and Barbara Ustanko, ELS, for editorial assistance with this manuscript.
Author Contributions
M.V. conceived the overall research project. M.V., K.S. and D.S. conceived the detailed research plan. M.V. and
K.S. directed the mathematical component of the paper. M.V. and R.B. directed the computational component
of the paper. M.Y. and K.I. performed the details of the mathematical research. R.S., M.F. and R.B. performed the
details of the computational component. M.V., K.S. wrote the main manuscript text; M.V., R.B. and R.S. wrote the
numerical methods; M.V., K.S., K.I. wrote the mathematical methods and proofs. R.S., K.I., K.S., M.F., M.V. and
M.Y. prepared gures for publication. All authors reviewed the manuscript.
Additional Information
Supplementary information accompanies this paper at https://doi.org/10.1038/s41598-017-12172-2.
Competing Interests: e authors declare that they have no competing interests.
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