a r X i v : 0 8 0 7 . 0 3 0 4 v 2 [ c o n d m a t . s o f t ] 2 5 M a r 2 0 0 9 Universality in the diffusion of knots Naoko Kanaeda ∗ and Tetsuo Deguchi † Department of Physics, Graduate School of Humanities and Sciences, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan (Dated: March 25, 2009) We have evalu ated a univ ersal ratio between dif fusion con stants of the ring polymer with a given knot Kand a linear polymer with the same molecular weight in solution through the Brownian dynamics under hydrodynamic interac tion. The ratio is found to be cons tant with respe ct to the numbe r of monomer s, N, and hence the estimat e at some Nshould be valid practically over a wide range ofNfor various polymer models. Interes tingly , the ratio is determined by the average crossing number (NAC) of an ideal conformation of knotted curve K, i.e. that of the ideal knot. The NAC of ideal knots should therefore be fundamental in the dynamics of knots. PACS numbers: 82.35.Lr,05.40.Fb,05.20.-y I. INTRODUCTION Novel knotted structures of polymers have recently been found in various research fields such as DNA, proteins and synthetic polymers. 1,2,3,4 The topology of a ring polymer is conserved under thermal fluctuations in solution and repre- sented by a knot. 5,6,7,8,9 Topological constraints may lead to nontrivial statistical mechanical and dynamical properties ofring polymers 6,10,11,12,13,14,15,16,17,18,19 . Recent progress in experiments of ring polymers should be quite remarkabl e. Dif fusio n constant s of linea r, relaxed circu lar and super coile d DNAs hav e been measured quite accurately. 20 Here the DNA double lelices are unknotted. Fur- thermore, hydrody namicradius of circu lar DNA has a lso been measured. 21 Ring polymers of large molecular weights are synthesized not only quite effectively 22 but also with small dispersions and of high purity. 23,24 Circular DNAs with vari- ous knot types are derived, and they are separated into knot- ted species by gel electrophoresis. 25 We should remark that synthetic ring polymers with nontrivial knots have not been synthesized and separated experimentally, yet. However, it is highly expected that ring polymers of nontrivial knot types should be synthesized and their topological effects will be confirmed experimentally in near future. In the paper we discuss diffusion constant D K of a ring polymer with fixed topology Kin good solution for various knot types. We evaluate it numerically via the Brownian dy- namics with hydrodynamic interaction in which bond cross- ing is effectively prohibited through the finite extensible non- linear elongational (FENE) potential. 26 We evaluate diffusion constant D L of a linear polymer with the same molecular weight , and derive ratio D K /D L . The ra tio should corre- sponds to a universal amplitude ratio of critical phenomena and play a significant role in the dynamics of knotted ring polyme rs. Accord ing to the renor mali zati on group arguments , ratio D K /D L should be unive rsa l if the number of monome rs, N, is large enough. 27,28,29 The ratio D K /D L may have some experimental applica- tions. Ring polymers of different knot types can be separated experimentally with respect to their topologies by making use of the difference among the sedimentation coefficients, which can be calculated from the diffusion constants. 32 Here we re- mark that the diffusion constant of a ring polymer under no topological constraint , D R , and that of the corresponding linear polymer has been numerically evaluated, and the ratio C= D R /D L has been studied. 26,30,31 Through simulation we find that ratio D K /D L is approxi- mately constant with respect to Nfor variou s knots. Thus, ifwe evaluate ratio D K /D L at some value ofN, it is practically valid for other values ofN. We can therefore predict the dif- fusion constant D K of a polymer model at some val ue ofN, multiplying the ratio D K /D L by the estimate ofD L . He re we remark that the value ofD L may depend on the number Nand on some details of the model. 33,34 Furthermore, we show numerically that ratio D K /D L is a linear function of the average crossing number ( NAC) ofthe ideal knot ofK, an ideal configuration of knotted curve K, which will be defined sho rtly . Since the ratio D K /D L is almost independent ofN, it follows that the linear fitting formula should be valid practically in a wide range of finite values ofN. Thus, the ideal knot of a knotted curve Kshould play a fundamental role in the dynamics of finite-size knotted ring polymers in solution. Let us introduce the ideal knot, briefly. For a given knot Kit is given by the trajectory that allows maximal radial expan- sion of a virtual tube of uniform diameter centered around the axial trajectory of the knot K. 35,36 We define the NACof a knotted curve as follows: We take its projection onto a plane, and enumerate the number of crossings in the knot diagram on the plane. Then, we consider a large number of projections onto planes whose normal vectors are uniformly distributed on the sphere of unit radius, and take the average of the cross- ing number (NAC) over all the normal directions. The paper consist s of the follo wig: In section II, the sim- ulat ion method is expla ined. In section III, we present the estimates of the diffusion constant of a ring polymer in solu- tion of knot type Kfor various knot ty pes. Then, we show numerically that the graph ofD K /D 0 is almost independent ofN, and also that ratio D K /D L is fitted by a linear function ofNACof the ideal knot ofK. We also discuss the simula- tion result in terms of the ratio of equivalent radii, 37 a G /a T, which corresponds to the universal ratio of the radius of gyra- tion to the hydrodynamic radius. 33 We shall define the equiva- lent radii explicitly in section III. Finally, we give conclusion in section IV. Throughout the paper, we emplo y the symbols of knots fol-
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8/3/2019 Naoko Kanaeda and Tetsuo Deguchi- Universality in the diffusion of knots
a r X i v : 0 8 0 7 . 0 3 0 4 v 2 [ c o n d - m a t . s o
f t ] 2 5 M a r 2 0 0 9
Universality in the diffusion of knots
Naoko Kanaeda∗ and Tetsuo Deguchi†
Department of Physics, Graduate School of Humanities and Sciences,
Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan
(Dated: March 25, 2009)
We have evaluated a universal ratio between diffusion constants of the ring polymer with a given knotK and a
linear polymer with the same molecular weight in solution through the Brownian dynamics under hydrodynamic
interaction. The ratio is found to be constant with respect to the number of monomers, N , and hence the estimateat some N should be valid practically over a wide range of N for various polymer models. Interestingly, the
ratio is determined by the average crossing number (N AC) of an ideal conformation of knotted curve K , i.e.
that of the ideal knot. The N AC of ideal knots should therefore be fundamental in the dynamics of knots.
PACS numbers: 82.35.Lr,05.40.Fb,05.20.-y
I. INTRODUCTION
Novel knotted structures of polymers have recently been
found in various research fields such as DNA, proteins and
synthetic polymers.1,2,3,4 The topology of a ring polymer is
conserved under thermal fluctuations in solution and repre-
sented by a knot.5,6,7,8,9 Topological constraints may lead tonontrivial statistical mechanical and dynamical properties of
ring polymers6,10,11,12,13,14,15,16,17,18,19.
Recent progress in experiments of ring polymers should
be quite remarkable. Diffusion constants of linear, relaxed
circular and supercoiled DNAs have been measured quite
accurately.20 Here the DNA double lelices are unknotted. Fur-
thermore, hydrodynamicradius of circular DNA has also been
measured.21 Ring polymers of large molecular weights are
synthesized not only quite effectively22 but also with small
dispersions and of high purity.23,24 Circular DNAs with vari-
ous knot types are derived, and they are separated into knot-
ted species by gel electrophoresis.25 We should remark that
synthetic ring polymers with nontrivial knots have not beensynthesized and separated experimentally, yet. However, it is
highly expected that ring polymers of nontrivial knot types
should be synthesized and their topological effects will be
confirmed experimentally in near future.
In the paper we discuss diffusion constant DK of a ring
polymer with fixed topology K in good solution for various
knot types. We evaluate it numerically via the Brownian dy-
namics with hydrodynamic interaction in which bond cross-
ing is effectively prohibited through the finite extensible non-
linear elongational (FENE) potential.26 We evaluate diffusion
constant DL of a linear polymer with the same molecular
weight, and derive ratio DK/DL. The ratio should corre-
sponds to a universal amplitude ratio of critical phenomena
and play a significant role in the dynamics of knotted ring
polymers. According to the renormalization group arguments,
ratioDK/DL should be universal if the number of monomers,
N , is large enough.27,28,29
The ratio DK/DL may have some experimental applica-
tions. Ring polymers of different knot types can be separated
experimentally with respect to their topologies by making use
of the difference among the sedimentation coefficients, which
can be calculated from the diffusion constants.32 Here we re-
mark that the diffusion constant of a ring polymer under no
topological constraint , DR, and that of the corresponding
linear polymer has been numerically evaluated, and the ratioC = DR/DL has been studied.26,30,31
Through simulation we find that ratio DK/DL is approxi-
mately constant with respect to N for various knots. Thus, if
we evaluate ratio DK/DL at some value of N , it is practically
valid for other values of N . We can therefore predict the dif-fusion constant DK of a polymer model at some value of N ,multiplying the ratio DK/DL by the estimate of DL. Here
we remark that the value of DL may depend on the number N and on some details of the model.33,34
Furthermore, we show numerically that ratio DK/DL is
a linear function of the average crossing number (N AC ) of
the ideal knot of K , an ideal configuration of knotted curve
K , which will be defined shortly. Since the ratio DK/DL
is almost independent of N , it follows that the linear fitting
formula should be valid practically in a wide range of finite
values of N . Thus, the ideal knot of a knotted curve K should
play a fundamental role in the dynamics of finite-size knotted
ring polymers in solution.Let us introduce the ideal knot, briefly. For a given knot K it is given by the trajectory that allows maximal radial expan-
sion of a virtual tube of uniform diameter centered around the
axial trajectory of the knot K .35,36 We define the N AC of a
knotted curve as follows: We take its projection onto a plane,
and enumerate the number of crossings in the knot diagram
on the plane. Then, we consider a large number of projections
onto planes whose normal vectors are uniformly distributed
on the sphere of unit radius, and take the average of the cross-
ing number (N AC ) over all the normal directions.
The paper consists of the followig: In section II, the sim-
ulation method is explained. In section III, we present the
estimates of the diffusion constant of a ring polymer in solu-
tion of knot type K for various knot types. Then, we show
numerically that the graph of DK/D0 is almost independent
of N , and also that ratio DK/DL is fitted by a linear function
of N AC of the ideal knot of K . We also discuss the simula-
tion result in terms of the ratio of equivalent radii, 37 aG/aT ,which corresponds to the universal ratio of the radius of gyra-
tion to the hydrodynamic radius.33 We shall define the equiva-
lent radii explicitly in section III. Finally, we give conclusion
in section IV.
Throughout the paper, we employ the symbols of knots fol-
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FIG. 2: Diffusion constants of linear and knotted ring chains with
knots 0, 31, 41, 51, 61 and 71, versus N . Fitted by D = aN −ν(1 +bN −∆) with the following best estimates: For a linear chain, a =0.90 ± 0.23, ν = 0.53 ± 0.06, b = 0.51 ± 0.93, ∆ = 1.14 ± 2.39,
χ2 = 17; for the trivial knot (0), a = 1.03± 1.11, ν = 0.55± 0.18,
b = 0.14±0.78, ∆ = 0.60±6.09, χ2 = 28; for the trefoil knot (31)
a = 1.00±3.87, ν = 0.52±0.67, b = 1.18±1.12, ∆ = 0.77±6.09,
χ2 = 27.
0
0. 5
1
1. 5
2
10 15 20 25 30 35 40 45 50
D
/ D
N
3
0
1
FIG. 3: Ratio D31/D0 of diffusion constants for the trefoil knot (31)
and the trivial knot (0) versus the number of segments N . Fitting
curve is given by D31/D0 = a(1 +bN −c), where a = 1.07±0.64,
b = 0.25 ± 0.39, and c = 0.39 ± 3.29 with χ2 = 6.
8/3/2019 Naoko Kanaeda and Tetsuo Deguchi- Universality in the diffusion of knots
FIG. 4: Ratio D41/D0 of diffusion constants for the figure-eight
knot (41) and the trivial knot (0) versus the number of segments N .Fitting curve is given byD41/D0 = a(1+bN −c) where a = 1.02±0.56, b = 1.76 ± 8.26, and c = 0.70 ± 2.58 with χ2 = 0.03.
1. 1
1.15
1. 2
1.25
1. 3
1.35
1. 4
1.45
1. 5
4 6 8 10 12
D / D
K
L
FIG. 5: DK/DL versus the average crossing number (N AC) of ideal
knot K for N = 45: The data are approximated by DK/DL =a + bN AC where a = 1.11 ± 0.02 and b = 0.0215 ± 0.0003 with
χ2 = 2.
8/3/2019 Naoko Kanaeda and Tetsuo Deguchi- Universality in the diffusion of knots