Topological interactions between ring polymers: Implications for chromatin loops

1.2 MB·English

by  Manfred Bohn

Topological interactions between ring polymers: Implications for chromatin
loops
Manfred Bohna) and Dieter W. Heermannb)
Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19, D-69120 Heidelberg,
Germany
(Dated: 24 January 2010)
Chromatin looping is a major epigenetic regulatory mechanism in higher eukaryotes. Besides its role in
transcriptional regulation, chromatin loops have been proposed to play a pivotal role in the segregation of
entirechromosomes. Thedetailedtopologicalandentropicforcesbetweenloopsstillremainelusive. Here,we
quantitatively determine the potential of mean force between the centers of mass of two ring polymers, i.e.
loops. We find that the transition from a linear to a ring polymer induces a strong increase in the entropic
0 repulsionbetweenthesetwopolymers. Ontop,topologicalinteractionssuchasthenon-catenationconstraint
1 further reduce the number of accessible conformations of close-by ring polymers by about 50%, resulting in
0 an additional effective repulsion. Furthermore, the transition from linear to ring polymers displays changes
2
in the conformational and structural properties of the system. In fact, ring polymers adopt a markedly more

n ordered and aligned state than linear ones. The forces and accompanying changes in shape and alignment
a between ring polymers suggest an important regulatory function of such a topology in biopolymers. We
J
conjecture that dynamic loop formation in chromatin might act as a versatile control mechanism regulating

4
and maintaining different local states of compaction and order.
2

]
t I. INTRODUCTION basepairs (Mb)3. In principle, active ATP-consuming
f
o mechanisms might be responsible for maintaining segre-
s Ring polymers are abundantly found in nature, com- gation. RosaandEveraers10 proposedthatchromosomes
t. monexamplesrangefromtheDNAofbacteriatocertain are aggregating into distinct compartments because of
a
m virus types1, mitochondrial DNA in higher eukaryotes theirlargerelaxationtime,makingitimpossibleforthem
andproteins2. Thesefindingssuggestastructuraladvan- to intermingle during the time of one cell cycle. How-
- ever, this study does not explain segregation of smaller
d tageofringpolymersoverlinearones. Thegenomiccon-
regions within chromosomes. Vettorel and co-workers11
n tent of higher eukaryotes inside the interphase nucleus,
o however, has a much more complex organization. Com- suggested that the collapsed state might be responsible
c paction is accomplished on several scales, the detailed for segregation: Polymer gels, where single polymers are
[ in a collapsed state, show no reptation, therefore inter-
folding pathways being closely related to genome func-
1 tion3,4. Nowadays, abundant experimental and theoreti- minglingbetweendifferentpolymersbecomespractically
v cal evidence suggests that chromatin looping acts as an impossible. Recently, evidence from polymer models has
6 indicatedthatchromatinloopsforcesuchacompartmen-
importantlinkbetweenchromatinorganizationandfunc-
4 talizationalreadybyvirtueofpurelyentropicforces12,13.
tion. The existence of chromatin loops on scales from a
2 fewkilobasepairs(kb)uptoseveralmegabasepairs(Mb) Whilethemeansquaredradiusofgyration R2 ofiso-
4 g
. hasbeenevaluatedby3C/4C/5Cexperiments5,6. Several latedringpolymersandasystemoftwocatenatedrings14
1 (cid:10) (cid:11)
mechanisms for looping have been proposed: Amongst displays a scaling similar to that of a self-avoiding walk,
0
others, it was suggested that these loops arise when dif- where R2 ∼N2ν withν ≈0.58815,non-catenatedrings
0 g
ferent genes and regulatory elements co-localize in tran- in a melt become compact adopting a scaling exponent
1
: scription hubs or factories7, where a high concentration of ν =(cid:10)1/3(cid:11)11,17. In fact, chromatin organization is not
v
of polymerases engages transcription. consistent with a simple melt of ring polymers. Scaling
i
X Inconsistent with theoretical predictions for solutions exponentsfoundinexperimentsdifferalot,rangingfrom
r of linear polymers, chromosomes arrange such that they ν ≈0.1−0.218 to ν =0.519, the level of compaction de-
a occupy distinct territories9, the overlap volume being pending on gene activity20. Dynamic chromatin loops,
quite small. A similar effect has been found even for however, have been shown recently to provide a consis-
smallpartsofsinglechromosomesspanningseveralmega tent framework of chromatin organization12.
Thepurposeofthisstudyistodeepentheunderstand-
ing of the topological interactions ring polymers exert
on each other as well as the changes in their conforma-
a)ThefollowingarticlehasbeenacceptedbyTheJournalofChem- tional properties induced by such forces. While topo-
ical Physics. After it is published, it will be found at http: logical effects of rings have been considered in several
//jcp.aip.org.;Electronicmail: [email protected] studies16,21,22, the induced forces have not been quanti-
b)Institute for Molecular Biophysics, The Jackson Laboratory,
tatively assessed. A system of two ring polymers can be
Bar Harbor, ME; Interdisciplinary Center for Scientic Comput-
viewed as a toy system for the influence of loop forma-
ing (IWR), University of Heidelberg, Im Neuenheimer Feld 368,
D-69120Heidelberg,Germany tiononchromatinfolding. Whilethebiologicalsystemis
2
beyond doubt much more complicated – multiple loops For the problem of ring polymers investigated here, the
and the dense system of polymers playing a dominant interactionpotentialV({rα})isgivenbyanexcludedvol-
i
role – the effect of topological forces and the advantage ume term and eventually by a term describing the topo-
oftheringshapeforbiopolymerscanbehighlightedbest logical constraints.
by studying it apart from other influences. We evaluate Thepotentialofmeanforcecanbegainedviathepar-
the potential of mean force between two ring polymers titionsumofthesystemconstrainedtoafixedcenter-of-
dependent on their mutual topological state to derive a mass,
quantitativemeasurefortheentropicrepulsiontworings
orloopsexertoneachother. Therefore,MonteCarlosim- 1 N 1 N
Z (R)=Tr exp(−βH)δ r(1)− r(2)−R
ulationsofringpolymerswithlengthsofuptoN =2048 c N i N i
(cid:34) (cid:32) (cid:33)(cid:35)
monomers are conducted. We separate the entropic ef- (cid:88)i=1 (cid:88)i=1
fects due to excluded volume interactions from the topo- The trace denotes the integral over all momenta and
logical interactions arising from the non-catenation con- monomer coordinates.
straint. The ring polymers’ conformational properties The effective potential is now defined by the relation
and their mutual alignment subjected to the topological
andentropicforcesareinvestigated. Infact,wefindthat exp(−βU (R))=Z (R)
eff c
ring polymers display a markedly stronger repulsion and
adoptamoreorderedstatecomparedtolinearpolymers. For purposes of calculation, however, the following rep-
The findings indicate a natural benefit of rings over lin- resentation24,25 of the effective potential is more useful
ear ones, leading to the conjecture that chromatin loops
Z (R)
not only facilitate entropy-driven segregation of chromo- U (R)=−k T ln c (1)
eff B Z2
somesorintra-chromosomalregions,butactasaversatile 1
control mechanism regulating and maintaining different
whereZ isthepartitionsumofasinglepolymer. Equiv-
states of compaction and order. 1
alently, Z2 may be imagined as the partiton sum of a
The paper is organized as follows: In section II we 1
two-chain system whose centers of mass are infinitely far
revisit the theory behind the effective potential, present
apart.
detailsofthesimulationalmethodandthenumericalcal-
culations of the effective potential. In section IIIA we
calculate the potential of mean force for linear polymers
B. Monte-Carlo algorithm
and review scaling theories. We turn to ring polymers
in section IIIB and study the effect of topological inter-
Thegenerationofringpolymersisaccomplishedusing
actions in section IIIC. The following two sections are
a well-tested lattice Monte Carlo algorithm, the bond
dedicatedtotheinvestigationofhowthedimensionsand
fluctuation model26,27. Simulations are set up on a cu-
shape of a ring changes when bringing two rings into
bic lattice, where each monomer of the ring resides in
contact. Finally, in section IIIF, we study the mutual
the center of a unit cube and occupies all its eight ver-
alignment of rings considering the effect of topological
tices. Each monomer is connected by a bond to its two
constraints.
neighborsalongthecontourofthepolymer,ensuringthe
connectivity of the chain. The lattice constant is taken
to be unity and all measures of length in the following
II. SIMULATIONS AND METHODS
aregiveninmultiplesofthelatticeconstant. Thelength
of the bonds between two neighboring monomers are al-
A. Effective potentials
lowed to fluctuate, but constrained to a set B of 108
vectors27. OneMonteCarlotrialmoveencompassesran-
To investigate the potential of mean force exerted be-
domly selecting one monomer and proposing a move to
tween the centers of mass of ring polymers, we use the
one of its nearest neighbors on the cubic lattice. Using
concept of the effective potential. The method has been
these local moves combined with the constrained set of
used for a variety of polymer systems, and for some even
bondvectorsB,ithasbeenshownthatnobond-crossings
analytical approximations have been found 23,24. A rec- can occur during the simulation run27, thus the topolog-
ommendable review is given in Ref.25.
ical state of the rings is preserved while propagating the
Let the monomer positions and momenta of the two
system. Excluded volume interactions are taken into ac-
polymers be denoted by rα and pα (i = 1,...,N,α =
i i countbypreventingalatticesitetobeoccupiedbymore
1,2), respectively. α denotes the polymer chain index
than one monomer.
andiindexessubsequentlythemonomersofthepolymer.
Isolated ring polymers with chain lengths between
The properties of the system are given by the complete
N =32andN =2048aresimulated. Asonlylocalmoves
Hamiltonian
are applied, consecutive conformations are highly corre-
N pα2 lated. To determine the relaxation time of a ring poly-
H({pα},{rα})= i +V({rα}) mer, we evaluated the autocorrelation function C(t)28 of
i i 2m i the squared radius of gyration A(t) = R2(t) and from
α=1,2i=1 g
(cid:88) (cid:88)
3
a center-of-mass distance r is then given by
K W (r)
U (r)=−k T ln i=1 i
eff B K
(cid:80)
The standard error of the effective potential is deter-
mined by randomly subdividing the K two-chain con-
formations into M = 50 smaller subsets and calculating
the effective potential U (r) for these subsets23. The
m
standard error is then calculated by
M
1
∆U (r)= (U (r)−U (r))2
FIG.1. Illustrationofdifferenttopologicalstatesoftworing eff (cid:118)M m eff
(cid:117) m=1
polymers. Theleftfigureshowsanon-catenatedconformation (cid:117) (cid:88)
(cid:116)
with Gauss linking number Φ=0. On the right, a catenated In a first step, only taking into account excluded vol-
conformation is displayed having a linking number of Φ=1.
ume interactions, the statistical weight W (r) of a two-
i
chain conformation is set to W =0 in case the excluded
i
volume condition is violated, i.e. two monomers occupy
C(t) we estimated the integrated autocorrelation time τ
the same lattice site, and W =1 in case it is not.
i
usingSokal’swindowingprocedure14,29. Twosubsequent
However, care has to be taken using the bond fluc-
conformationsareconsideredindependentafter5τ Monte
tuation model. During the simulation run it is ensured
Carlo steps (MCS). In the following we define one MCS
that no bond crossings can occur; this, however, is not
asN trialmoves, whereN isthechainlengthandthere-
guaranteedifwejustshifttwoconformationsinsideeach
fore the total number of monomers in the system. Thus,
other. There are two possible combinations of bond vec-
in one MCS each monomer is proposed on average one
tors, where the excluded volume condition is not vio-
move. For each chain length at least 10000 independent
lated, but where the bonds come into contact. Consider
conformations were produced.
the two bonds spanned by the following four monomers
Simulations are performed on a lattice with periodic {(0,0,0) → (3,1,0),(1,2,0) → (2,−1,0)}. Both bond
boundary conditions, but the algorithm keeps track of vectors (3,1,0) ∈ B and (1,−3,0) ∈ B are valid and the
unfolded coordinates, such that the entire configuration excludedvolumeconditionissatisfied. Thesecondprob-
space can be explored. The linear lattice size L is cho- lematic case is {(0,0,1) → (2,2,0),(0,2,0) → (2,0,1)}.
sen large enough, i.e. L (cid:29) R2 1/2, so that monomers As can be easily seen, the participating beads are stuck
g
do not come into contact in the lattice space by peri- in these positions, there is no valid move for any of the
odic backfolding of coordinat(cid:10)es, o(cid:11)therwise biased confor- beadsusingtherestrictionsofbondvectorsthealgorithm
mations would result. Lattice sizes used vary between is subjected to. Therefore, during a simulation run such
L=256 up to L=800, depending on chain length N. a situation can never happen, assuming the start config-
uration is chosen properly. Here, we test each two-chain
conformation, after positioning them with their centers
of mass a given distance r apart, on such a situation. If
C. Calculation of the effective potential
abondcrossingoccurs, aweightofW =0isassignedto
i
this conformation.
Singleringconformationsareusedtocalculatetheav- In this study, we are furthermore interested in the
erageinteractionenergybetweentwopolymersbymeans topological state of a two-ring conformation. We cal-
of equation (1). This procedure is well-established and culate the weight factor differently whether we consider
hasbeenappliedinseveralstudies23,24. Atfirst,thesetof a preserved topological state or not. In the case that we
sampled conformations C is split into two parts of equal force a certain topological state, we have to add another
size. Then, we randomly pick two conformations, one criterion for accepting a two-chain conformation (i.e. as-
fromeachsubset. Thesetworingconformationsarethen signing a weight W (cid:54)=0). Two rings that are positioned
i
positioned such that their centers of mass have a certain atacenter-of-massdistancer mightbenon-catenatedor
distance r, the angular positioning in doing so is cho- catenated. Furthermore, the degree of catenation might
sen randomly. In order to ensure all monomers residing vary(seeillustrationsinFig.1). Wedeterminethetopo-
on lattice sites after shifting, a maximum distance in- logical state by a topological invariant: the “Gauss link-
accuracy of δr ≈ 0.87 lattice units is accepted. In the ing number”30. This invariant has been used in analyti-
next step, the interaction energy Ei(r) of the two-ring cal studies on interlinked rings to keep track of a certain
conformation and the corresponding statistical weight topology30,31. It is defined by
W (r)=exp(−E /k T) are calculated.
i i B
Repeating this operation for a number K of randomly Φ(C ,C )= 1 (cid:104)dr1×dr2,r1−r2(cid:105) (2)
selected pairs of conformations, the effective potential at 1 2 4π |r −r |3
(cid:73)C1(cid:73)C2 1 2
4
Theclosedlineintegralsareevaluatedalongthecontours a decrease with growing chain length, inconsistent with
of the two rings, denoted by C (k = 1,2). The vector the mean-field argument by Flory32 but consistent with
k
function r = r (s) denotes the three-dimensional coor- eq. (3). The effective potential at full overlap is shown
k k
dinates of the ring polymers, parameterized e.g. by the in a scaling plot (cf. Ref.36) in Fig. 2B. Data points are
contour length s. on a straight line as expected, deviations are most likely
The Gauss link invariant has the advantage that it is due to lattice effects: When the two polymer coils are
easytoevaluate,althoughithasthedisadvantagethatit brought to a certain distance, the nearest lattice site is
is not one-to-one. The “Whitehead link”30 is one exam- chosen,suchthatthedistancecandeviatefromthetarget
ple for a two-ring conformation having the same linking distance r = 0. These deviations are more pronounced
number as non-catenated rings. However, for this link, the smaller N.
self-intersections of one ring are necessary, which do not
arise during the simulation runs. Therefore we consider
theGausslinkingnumberasanexcellentquantitytode- B. Rings have a stronger repulsion than linear polymers
termine the topological state. The result of the integral
isanintegerfornon-overlappingchains,especiallyΦ=0 Tostudytheeffectivepotentialbetweenringpolymers,
for non-catenated chains and Φ=1 for simple catenated we first neglect topological constraints between the two
chains (Fig. 1). We numerically evaluate this integral to rings when bringing them in proximity. However, each
classify the topological state of a two-ring conformation single ring obeys the topological constraint of unknot-
andassignaweightWi =0foralltwo-ringconformation tedness by virtue of the simulational method. Thus the
notinthetopologicalstateweareinterestedin. Thereby effectivepotentialisgivenbythelogarithmoftheproba-
weareabletocalculatetheeffectivepotentialseparately bility that the ring conformations do not occupy at least
for any given topological state. one lattice site in common.
Figure3AshowstheeffectivepotentialU (r)forring
ring
polymers where mutual topological constraints are ne-
III. RESULTS
glected, i.e. the linking number between the two ring
polymers is arbitrary. The effective potential again is
A. The effective potential of self-avoiding walks dependent on chain length N, however, differences be-
come smaller for large chain lengths. The mean field
We first recapitulate the results for the effective po- argumentfortheeffectivepotentialatfulloverlapestab-
tential between the centers of mass of two self-avoiding lished in eq. (3) does not depend on the connectivity of
walks both to validate the algorithm used and to relate the chain. Indeed, it is not only valid for linear poly-
thestrengthandformofthepotentialforlinearpolymers mers, but also for rings. Figure 3B shows the effective
to their ring counterparts. potential at full overlap U (0) such that a linear ex-
ring
A simple mean field argument can be devised for the trapolation to N → ∞ becomes possible. Clearly, the
entropicrepulsionoftwopolymercoilswithexcludedvol- effective potential adopts a constant value in this limit.
ume at full overlap, i.e. with zero center-of-mass sepa- Interestingly, the repulsive interactions between two
ration. It has been first proposed by Flory and Krig- ring polymers are much stronger than for their lin-
baum32 and later been corrected in other studies33,34. ear counterparts of equal chain length N. The ratio
The overlap volume is given by V ∼ Rg3 where Rg is Uring(r)/USAW(r)isdisplayedinFig.4andhasavalueof
the radius of gyration of one polymer coil. A mean-field about 3 for center of mass (CM) separations below 2R .
g
argument yields the effective potential at full overlap34:
Scaling arguments33, which take into account the corre-
lation hole effect35, show that the probability p1 of one C. Topological constraints induce additional repulsion
monomer of chain A being in contact with any monomer
of B is given by p1 = (na3)3ν1−1. Thus the probability In the last section we have shown that the repulsive
that none of the monomers are overlapping is given by forcesactingbetweentworingpolymerswhosecentersof
p = (1−p1)N = (1− Nb(cid:48))N. The effective potential can mass are brought in close proximity are much stronger
then be calculated as the logarithm of this probability, than for linear self-avoiding walks, the potential being
b(cid:48) about 3 times larger. This rise in strength results from
U(r =0)=−Nln 1− (3) the increased density of monomers induced by the ring
N
(cid:18) (cid:19) structure. Here we focus on the additional effect topo-
InthelimitofinfinitechainlengthN →∞, theeffective logical constraints induce on the potential of mean force
potential at full overlap approaches a constant value in between the polymers. Much attention has been paid to
theorderofk T. Consequently,linearpolymercoilswith theinfluenceoftopologicalconstraintsonringpolymers.
B
excludedvolumehavearathersoftpotential,allowingfor The interaction between non-linked ring polymers with-
a high level of mutual penetration. out excluded volume has been investigated for chains of
Results for the self-avoiding walk are displayed in length up to N = 8021. In a recent study14 we have
Fig.2. TheeffectivepotentialU (r)inFig.2Areveals analyzed the behavior of two catenated rings. It turned
SAW
5
A B
2.5 1.02
N=64
N=128 1.01
N=256
1
) 2 N=384
r
(W N=512 0.99
A N=1024
US N) 0.98
tial 1.5 /0) 0.97
n (
e U
ot − 0.96
p 1 (
e xp 0.95
v e
cti 0.94
e
ff 0.5
e 0.93
0.92
0 0.91
0 0.5 1 1.5 2 2.5 3 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
r/R 1/N
g
FIG. 2. Effective potential acting between the centers of mass of two self-avoiding walks. A. This panel displays the effective
potential U (r) for different chain lengths. The x-axis is scaled by the root mean squared radius of gyration R of isolated
SAW g
chains. Errors are smaller than the point size and therefore not shown. The effective potential is dependent on chain length.
B. Scaling plot of the effective potential at full overlap. The effective potential U (0) reaches a finite value for large chain
SAW
length as expected from scaling theory (cf. equation (3)).
A B
6 1.05
N=64
N=128
5 N=256
) N=384 1
r
( N=512
g
Urin 4 N=1024 )
N=1536 N
tial N=2048 /0) 0.95
en 3 U(
ot −
p ( 0.9
tive 2 exp
c
e
ff
e 0.85
1
0 0.8
0 0.5 1 1.5 2 2.5 3 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
r/R 1/N
g
FIG. 3. Effective potential U (r) of ring polymers disregarding their mutual topological state. A. This panel displays the
ring
effective potential in units of k T for several chain lengths. Errorbars are not shown as they are smaller than the point size.
B
For reasons of comparison, the x-axis is scaled with the radius of gyration of the corresponding isolated ring polymer. B. The
scaling law for the effective potential at full overlap follows the theory from eq. (3) allowing for a linear extrapolation to the
infinite chain limit when plotting e−U(0)/N vs 1/N. For the extrapolation, only chain lengths larger than N =256 are used.
out that while the dimensions of catenated rings show N1/3. The dominant entropic force driving this com-
thesamescalingbehaviorthantheirlinearcounterparts, paction stems from the topological constraint of non-
the shape changes significantly. The topological states catenation. To our knowledge, a quantitative study of
of circular DNA have been analyzed in various publica- this topological potential has not been considered yet.
tions37,38. The studies on non-catenated and unknot- In the framework of effective interactions, the topologi-
ted ring polymers in the melt revealed that their di- cal non-catenation constraint can be included easily. As
mensions change dramatically11,16,17: They behave like explained in the section IIC, we determine the topologi-
compact polymers with a radius of gyration scaling with calstateofatwo-ringconformationbymeansofitsGauss
6
A
3.5 6
N=64
) N=128
3 r 5
( N=256
g
r() 2.5 0Urin 4 NN==358142
SAW tial N=1024
r/U() 2 N=64 poten 3 N=2048
ng 1.5 N=128 e 2
Uri N=256 tiv
N=384 c
1 e 1
N=512 ff
e
N=1024
0.5 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
distance r/Rg distance r/Rg
B
0.9
FIG.4. Comparisonoftheeffectivepotentialsforringpoly-
mers and self-avoiding walks. The ratio U (r)/U (r) is 0.8
ring SAW
plotted against the center-of-mass distance r. The x-axis is 0.7 0.65
scaledbytherootmeansquaredradiusofgyrationR ofiso-
g 0.6
lated chains for reasons of comparison. 0.64
r) 0.5
(
Utop 0.4 0.63
linking number Φ (eq. (2)). A Gauss linking number of N=1536 N=256
0.3
zero indicates the non-catenation of the two ring poly-
mers. To obtain the potential of mean force under the 0.2
non-catenation constraint, we calculate the fraction p of 0.1
two-ring conformations with both zero linking number
0
and fulfilled excluded volume constraint. The effective 0 0.5 1 1.5 2 2.5 3
potential is then given by U0 (r) = −lnp (the index
ring distance r/Rg
0 indicates the Gauss linking number Φ = 0). The re-
sulting potential Ur0ing(r) is shown in Fig. 5A. We find FIG. 5. A. The effective potential Ur0ing(r) for non-
pronounced deviations from a Gaussian shape for small catenated rings. B. The topological potential U (r) for
top
CM separations r (cid:46)1Rg. non-catenated rings is obtained as the difference between
From the effective potential U0 (r) of non-catenated U0 (r) and U (r). It displays the potential arising purely
ring ring ring
rings and U (r) of rings without topological con- from the topological non-catenation constraint. The inset
ring
straints,thetopologicalpotentialU (r),i.e. thepartof shows the maximum of the topological potential depending
top
U0 (r) stemming purely from the non-catenation con- on the chain length N. The maxima are obtained by a fit of
ring U (r)toathird-orderpolynomialintheinterval[0.3:0.7].
straint, can be obtained, ring
U (r)=U0 (r)−U (r)
top ring ring
(see inset of Fig. 5B). In numbers, this means that only
The topological potential is shown in Fig. 5B. It has a about 48% (exp(−0.65) ≈ 0.52) of all two-ring confor-
maximum at about r/R ≈ 0.55, a value which basi- mations with fixed center-of-mass distance which would
g
cally does not change with chain length N. For larger be allowed with respect to excluded volume are actually
CM separations U (r) drops to zero, as less and less allowed with respect to topological constraints. In other
top
two-ring conformations will be catenated when increas- words, only half of the two-ring conformations satisfying
ing r. There is also a decrease in the topological po- the excluded volume constraint are actually accessible
tential when going to small CM separations r < 0.5R . when the non-catenation constraint is considered. Thus,
g
This decrease becomes smaller for larger chain length the non-catenation constraint decreases the number of
N. The reason for this behavior is that for small r accessible conformations at small CM separations con-
(andsmallN),two-ringconformationsnotsatisfyingthe siderably.
non-catenation constraint, often do not satisfy the ex- Interestingly,thetopologicalinteractionisontheorder
cluded volume constraint either, such that the confor- of magnitude of 1k T. This value was used for a crude
B
mations are not counted for the purely topological in- estimateofthefreeenergyofringsinameltbyCatesand
teraction U . However, more interesting is the maxi- Deutsch39, wheretheyassumedthatthefreeenergycost
top
mumrepulsionexertedbythenon-catenationconstraint. for a contact between two rings is about 1k T, hence
B
We find that the topological potential is asymptotically in a three-dimensional melt the topological contribution
Umax ≈ 0.6 − 0.7 k T for (N → ∞) at r/R ≈ 0.55 to the free energy is F ≈ k TR3/N. Although highly
top B g B
7
A A
1.12 1.02
non-catenated
1.1 all rings
SAW c 1.01
n
(cid:11)
1.08 2g
R
Rg 1.06 / (cid:10) 1
Rr/()g 1.04 2Rgnt (cid:10)(cid:11)0.99
1.02
0.98
1
0 0.5 1 1.5 2 2.5 3
0.98 B CM separation s =r/Rg
0 0.5 1 1.5 2 2.5 3 3.5 4
1.02
c
B CM separation r/Rg 2Rgn (cid:11)1.01 ss==10
ν 1 (cid:10)
2N 0.9 / 1
/ nt
2Rg 0(cid:11).8 2Rg (cid:11)0.99
0(cid:10).7 (cid:10)0.98
100 1000 0 0.01
chain length N inverse chain length 1/N
FIG.6. A. Meansquaredradiusofgyrationforself-avoiding FIG.7. A. Ratiobetweenthemeansquaredradiiofgyration
walks (SAW), ring polymers without mutual topological in- of rings without mutual topological interactions (NT) and
teractions and non-catenated rings vs. distance between the non-catenated rings (NC) vs. scaled distance r/R between
g
centers of mass. Radii of gyration are scaled by the corre- the centers of mass. Results are shown for a chain of length
sponding values R of the isolated chains. B. Scaling of N =1024. B. The ratio is plotted against the inverse chain
g
the radius of gyration with chain length. The leading term lengthallowingforanextrapolationtotheinfinitechainlimit
N2ν of a self-avoiding walk (ν =0.588) is divided out to see N → ∞. Data is shown for two CM separations: s = 0 and
deviations from the linear chain behavior. s = 1 to analyze whether deviations remain in the limit of
large chain length.
simplistic, this scaling exponent ν = 2/5 was later re-
produced by simulational results16; more recent results,
however, suggest that this behavior is only a cross-over
to the behavior of compact lattice animals17. 0.306
0.304
0.302
D. Dimensions of two-ring conformations
0.3
(cid:11)
2d 0.298
When two polymers are brought closer and closer to- (cid:10)
/ 0.296
gether,loweringthedistancebetweenthecentersofmass, (cid:11)
2g 0.294
not only the repulsive interaction increases dramatically. R
(cid:10)0.292
Also the conformational properties of the polymers are
affected23 inaneffortofminimizingthefreeenergy. Here 0.29
non-catenated rings
we investigate how the dimensions of a two-ring confor- 0.288
all rings
mation change compared to a self-avoiding walk when 0.286
approaching isolated polymers. 0 0.5 1 1.5 2 2.5 3 3.5 4
Therearetwomeasuresofthedimensionsofthechain, CM separation r/Rg
which, however, exhibit the same scaling: the mean
squared radius of gyration R2 and the mean squared FIG. 8. Ratio between the mean squared radius of gyration
g
ringdiameter d2 (seeRef.14). Figure6Ashowstheroot andmeansquaredringdiametervscenter-of-massseparation
(cid:10) (cid:11) r. DataisshownforachainlengthofN =1024. Resultsare
mean squared radius of gyration R (r)= (cid:104)R (r)2(cid:105) for
(cid:10) (cid:11) g g shownfortwo-ringconformationswithoutmutualtopological
self-avoidingwalks(SAW),ringpolymerswithoutmutual
interactions (open circles) and non-catenated ring conforma-
(cid:112)
topologicalinteractions(arbitrarylinkingnumberΦ)and
tions (solid red circles).
non-catenated rings (linking number Φ = 0) in depen-
dence of the scaled center-of-mass separation s = r/R .
g
8
The data are scaled with the radius of gyration for iso- sion should yield a constant ratio. For a random walk,
lated polymers R = R (r = ∞) (i.e. the radius of gy- the ratio between the mean squared radius of gyration
g g
ration of a polymer at infinite distance away from the and the end-to-end distances is 1/6. For isolated ring
second one) for reasons of comparison. For small CM polymers,theratio R2 / d2 extrapolatesto0.3053(2)
g
separations (r < R ) the radius of gyration strongly in the asymptotic limit, while catenated rings display a
g
increases and its dimensions are more than 10% larger ratio of 0.2995(3)14(cid:10),22.(cid:11)Fo(cid:10)r t(cid:11)he case studied here, the
than in the isolated case. This is consistent with the re- ratio between the radius of gyration and ring diameter
sults for self-avoiding walks by Dautenhahn and Hall23. R2 / d2 changes significantly for different CM sepa-
g
The increase in dimension at separations r (cid:28) R origi- rations (see Fig. 8). From its isolated chain value the
g
(cid:10) (cid:11) (cid:10) (cid:11)
nates from the need of the chains to create space for the ratio decreases when bringing the centers of mass closer
monomers of the second chain. At intermediate separa- together. Extrapolation to infinite chain length N →∞
tions of about r/R ≈1.5−2, the radius of gyration at- showsthatthedifferencesremaineveninthislimit(Sup-
g
tainsasmallervalue,thusthepolymereffectivelyunder- plementary Figure 242). At full overlap (s = 0) the
goes compaction. Both the compaction at intermediate ratio extrapolates to R2 / d2 = 0.2874(3) for non-
g
separationsaswellastheswellingathighoverlapremain catenated rings and R2 / d2 = 0.2891(3) for rings
(cid:10) g (cid:11) (cid:10) (cid:11)
in the limit of infinite chain length. An extrapolation of without topological interactions.
the radius of gyration at full overlap (s=0) to N →∞ (cid:10) (cid:11) (cid:10) (cid:11)
shows that R (r = 0)/R → 1.109(1) for rings without
g g
topological interactions and Rg(r = 0)/Rg → 1.111(1) E. Shape of two-ring conformations
for non-catenated rings (see Supplementary Figure 140).
One might ask, whether this change in dimensionality A typical measure of the shape of a polymer coil is its
is accompanied by a change in the scaling law. We know gyration tensor43,44. It is defined by
from several studies that isolated ring polymers follow a
sνim≈il0a.r59sc1a4,l2i2n,g41l.awHoawselvineera,rnponol-ycamteernsateRdg2rin∼gsNi2nνawmheerlet Smn = N1 N rm(i)rn(i) (4)
(cid:10) (cid:11)
behave like compact polymers with a scaling exponent (cid:88)i=1
ν =1/311,17. Figure6Bshowsthescalingoftheradiusof Here r(i) is the coordinate vector of the ith monomer
gyrationwithchainlengthinalog-logplotfortwocenter-
and the subindex denotes its cartesian components. The
of-massseparations: r =0(fulloverlap,regionwherethe
matrix S is symmetric and positive semi-definite, thus it
chain is swollen) and r = 1.7R (region where the chain
g canbetransformedtoadiagonalmatrixwherethethree
is compacted). The leading order term N2ν of the self-
eigenvalues λ ≤ λ ≤ λ give the squared lengths of
avoidingwalkbehavior(ν =0.58815)isdividedout. The 1 2 3
the principal axes of gyration of the associated gyration
data adopt a constant value in the limit of large chain
ellipsoid. In fact, the sum of the eigenvalues gives the
length N independent of CM separation, indicating that squared radius of gyration R2 =λ +λ +λ .
g 1 2 3
thescalingbehaviorisindependentofseparation. Asthis
In a recent study14 on catenated rings we have found
also applies to rings with the non-catenation constraint,
thattheshapeofcatenatedringsdifferssignificantlyfrom
thecompactionfoundinapolymermeltofnon-catenated
theshapeofisolatedringpolymers. Thisshowsupinthe
ringsseemstorequireseveralchains,exertingforcesfrom
ratiosoftheaveragedeigenvalues(cid:104)λ (cid:105):(cid:104)λ (cid:105):(cid:104)λ (cid:105),which
all sides on the polymer coil. 3 2 1
are obtained from extrapolation to the asymptotic limit
Ourresultsshowthatdifferencesbetweentheradiusof
as 8.23(2) : 3.22(2) : 1 for isolated rings and 8.87(3) :
gyration R2 of rings without topological constraints
g nt 3.56(2):1 for catenated rings.
and Rg2 (cid:10)nc o(cid:11)f rings obeying non-catenation are rather We investigate how the shape of rings changes when
subtle. Theratiooftheradiiofgyration R2 / R2 approaching the centers of mass, both for rings with-
(cid:10) (cid:11) g nt g nc
with respect to the CM separation is shown in Fig. 7A out topological interactions as well as for conformations,
(cid:10) (cid:11) (cid:10) (cid:11)
for a chain length of N = 1024. For small separations which obey the non-catenation constraint. Figure 9A
r < 0.5R non-catenated rings are smaller, while for in- displays the results for a chain of length N =1024. The
g
termediate separations (r ∼ 1R ) they are larger. The polymers get strongly elongated for small center-of-mass
g
dependenceoftheseresultsonthechainlengthN isgiven separations s = r/R . This elongation at full overlap
g
in Fig. 7B for two different CM separations. The dif- s=0 remains in the limit of large chain length N →∞
ferences are asymptotically stable both in the regime of as is displayed in Fig. 9B. For non-catenated rings, the
smallaswellasintermediateseparations. Atfulloverlap effect of elongation is slightly more pronounced than for
(s = 0), non-catenated rings are larger in size by about rings, where topological effects are neglected. At full
0.8% in the asymptotic limit N → ∞, for intermediate overlap we find for non-catenated rings that the ratios
separations(s=1),ringswithouttopologicalconstraints (cid:104)λ (cid:105):(cid:104)λ (cid:105):(cid:104)λ (cid:105)areasymptotically11.05(8):4.32(3):1,
3 2 1
are larger by about 1.5%. Similar results are found for while for rings without mutual topological interactions
the mean square ring diameter (data not shown). we have 10.84(6):4.17(2):1.
Asisolatedringpolymersaswellasself-avoidingwalks While the analysis of the radius of gyration indicated
only have one length scale, different measures of dimen- that the spatial extend of rings becomes larger, we have
9
AAA 12 BBB
non-catenated rings i 13
1i all rings λ1 12
λ 10 h
/h /i 11
λ3i 8 λ3h 10
h 0 0.005 0.01 0.015 0.02
6 inverse chain length 1/N
i
1 6
λ i
h 4 1
/ λ
2i /h 5
λh 2 2i
λ
h 4
0 0.5 1 1.5 2 2.5 3 3.5 0 0.005 0.01 0.015 0.02
CM separation r/Rgyr inverse chain length 1/N
FIG.9. A.Ratiosbetweenthegyrationtensor’seigenvaluesλ ≤λ ≤λ inrelationtothecenterofmassdistances=r/R .
1 2 3 g
Data is shown for a system with chain length N = 1024 for two-ring conformations without mutual topological interactions
(open circles) and non-catenated ring conformations (solid red circles). B. Finite-size behavior of the ratios (cid:104)λ (cid:105)/(cid:104)λ (cid:105) and
3 1
(cid:104)λ (cid:105)/(cid:104)λ (cid:105) for a center-of-mass separation of s=0.
2 1
found here that this opening up does not lead to a com- each other. An extrapolation to the infinite chain length
plete mixing of the two rings, which would result in a regime at full overlap (s = 0) yields an average angle
more spherical structure; rather rings elongate strongly (cid:104)cosθ (cid:105) of 0.3543(7) for rings without topological inter-
1
in one direction. actions, 0.3492(6) for non-catenated rings and 0.3897(2)
for self-avoiding walks. Details of the extrapolation are
shown in the Supplementary Figure 347.
F. Orientation and Alignment of rings The angle (cid:104)cosθ (cid:105) (see Fig. 10B) shows a nearly ran-
2
domalignmentforCMseparationsofr ≈0. Forinterme-
diate r ≈1−2R , the alignment becomes pronouncedly
Toinvestigatetheinteractionsbetweenthetworingsat g
more perpendicular compared to a random alignment.
full overlap in more detail, we look at the mutual align-
Thiseffectismuchstrongerfornon-catenatedringsthan
ment and orientation. The alignment of the gyration
forringswithoutmutualtopologicalconstraints. Clearly,
ellipsoids of both rings with respect to each other can
ring polymers display a much stronger tendency to align
be investigated via two measures: Firstly, the average
non-randomly than self-avoiding walks. In the regime of
angle (cid:104)cosθ (cid:105) between the largest main axes of the el-
1
perpendicular alignment at s = 1.2 the asymptotic val-
lipsoids, secondly the average angle (cid:104)cosθ (cid:105) between the
2
ues of 0.3836(5) (rings without topological interactions),
vector connecting the centers of mass of both rings and
0.3661(4) (non-catenated rings) and 0.4327(3) (SAWs)
the largest main axis of one ellipsoid. Its behavior has
been observed for catenated rings14. It was found that are found for the average angle (cid:104)cosθ2(cid:105)47.
themainaxesoftheellipsoidstendtoalignperpendicular A recurrent motif in the analysis of shape, dimensions
forshortCMseparations,whileforlargeCMseparations, and orientation is a change in structure from polymers
the alignment becomes more and more parallel. at full overlap to polymers at intermediate separations
Theresultsfortwo-ringconformationswithoutmutual (r ≈ 1−2Rg). While they are strongly elongated and
topologicalconstraintsandnon-catenatedringconforma- aligned perpendicular at short separations, at intermedi-
tionsaswellasSAWsareshowninFig.10. Theblackline ate separations compaction and parallel alignment is ob-
at (cid:104)cosθ (cid:105) = 0.5 corresponds to the average angle for a served. Clearly, thisresultsfromatendencytominimize
i
randomorientationofthevectors(Notethattheangleθ theoverlapareabetweenbothrings. Atfulloverlap,this
i
canadoptonlyvaluesbetweenzeroand90degrees). The is accomplished best by a strong elongation and perpen-
averageangle(cid:104)cosθ (cid:105)betweenthegyrationtensorsmain dicular alignment (Fig. 11A). When rings are separated
1
axes (cf. Fig. 10A) displays a pronounced perpendicu- further apart, the gyration ellipsoids can avoid intermin-
lar orientation for small CM separations r < R . This gling by aligning in parallel (Fig. 11B). The restricted
g
effect is markedly stronger for ring polymers than for conformational space due to the presence of the other
linear ones. At intermediate separations of r ≈1−2R , ring results in the observed compaction of the radius of
g
however,aregimeisfoundwhereaslightlyparallelalign- gyration compared to isolated chains.
ment of the rings is preferred compared to a random ori- Both the topological constraints and the ring connec-
entation. For large center of mass separations, a ran- tivity of the chain have a strong influence on how the
dom orientation is approached; in this regime, the rings polymers are aligning when brought close together. In
are nearly independent and therefore do not influence general, theviewemergesthatringshaveamorealigned
10
AA center−of−mass distance s=0.3
0.54
0.52
0.5
0.48
i 0.46
1
θ
s 0.44
o
ch 0.42
0.4
0.38 non-catenated rings
center−of−mass
all rings
0.36 linear chain (SAW) distance s=1.5
0.34
0 0.5 1 1.5 2 2.5 3
FIG.11. Sampleconformationsofnon-catenatedringsforced
CM separation r/Rgyr toacertaincenter-of-massdistance. Theleft-handfiguredis-
BB playsaconformationoftwonon-catenatedringsatnearlyfull
0.52
overlap (s=r/R =0.3). The two rings markedly align per-
g
0.5 pendicularly. The right-hand figure shows a two-ring confor-
0.48 mation of non-catenated rings at separation s = r/Rg = 1.5
displaying a tendency for parallel alignment.
0.46
i
2 0.44
θ
s
co 0.42 conformation is analyzed by means of the Gauss linking
h
0.4 number.
We found that the effective potential at full overlap
0.38
adopts a constant value both for linear chains and ring
0.36
polymers(Figs.2and3). However,therepulsiveinterac-
0.34 tion between the centers of mass is about 3 times larger
0 0.5 1 1.5 2 2.5 3
for ring polymers than the effective potential for corre-
CM separation r/Rgyr sponding linear chains (Fig. 4) at small center-of-mass
separations r.
FIG.10. A.Averageangle(cid:104)cosθ (cid:105)betweenthelargestprin-
1 If the ring polymers have to stay in the fixed mutual
cipal axes of the gyration ellipsoids of a two-chain conforma-
topological state of non-catenation, the space of accessi-
tion in relation to its center of mass separations. The black
ble conformations is further reduced, thus the effective
linecorrespondstoarandomorientationofthetwoaxes. Re-
potential increases. We have evaluated the strength of
sults are shown for self-avoiding walks (grey triangles), non-
catenated rings (solid red circles) and rings without mutual these interactions – the topological potential Utop(r) –
interactions (open black circles). B. Average angle (cid:104)cosθ (cid:105) resulting purely from topological constraints. We find
2
between the vector connecting the centers of mass and the that the non-catenation constraint further increases the
largest principal axis of one polymer coil in relation of the total repulsive interaction by about 10%. The strength
CM separation. of the potential is of the order of 1k T at small separa-
B
tions r. The number of rejected conformations increases
by over 50% compared to only taking into account ex-
and ordered state than linear chains, an effect which is cluded volume interactions.
even amplified by the non-catenation constraint.
The analysis of conformational properties of polymers
brought close together reveals effects of the ring struc-
turebothonsizeandshape. Ringpolymersinproximity
IV. CONCLUSIONS become swollen (Fig. 6) and strongly elongated (Fig. 9)
compared to the situation of isolated or far apart ones.
In this study we investigated the effect of the non- While both effects are found for linear polymers, too,
catenation constraint on a system with two ring poly- the effects are more pronounced for their ring counter-
mers. Our main focus was on the quantitative analysis parts. Additional effects from the non-catenation con-
ofthestrengthoftheinteractionsbetweenringpolymers, straint concerning size and shape are visible, but small,
including their topological interactions. For this pur- indicating that the changes are mainly induced by the
pose, we sampled conformations of isolated rings using presence of additional material due to the rings being
a well-established Monte Carlo method. We evaluated more compact than linear polymers.
the potential of mean force using the idea of effective Remarkableeffectsoftheringstructurearefoundcon-
interactions25 following the methodintroduced by Daut- cerning the alignment of ring polymers in proximity.
enhahn and Hall23. The topological state of a two-ring There is a strong tendency to align perpendicular for