Effects of edible chitosan coating on quality and shelf life of sliced mango fruit

Structure and relaxation dynamics of

polymer knots

Pik-Yin Lai

_{, Yu-Jane Sheng}

_{, Heng-Kwong Tsao}

a_{Department of Physics and Center for Complex Systems, National Central University,}

Chung-li 320, Taiwan, ROC

b_{Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan, ROC} c_{Department of Chemical Engineering, National Central University, Chung-li 320, Taiwan, ROC}

Abstract

Monte Carlo simulations are performed to study the equilibrium structure and nonequilibrium dynamic relaxation processes of knotted polymers. We nd that topological complexity aects the static and dynamic behavior of knots in dierent ways to dierent extent. For statics, our results on the radii of gyration of knot polymers suggest that prime and two-factor composite knots belong to dierent groups, and we conrm that for knots in the same group, the average radius of gyration scales as Rg ∼ N3=5p−4=15 in good solvents, where N is the number of

monomers and p is the topological invariant representing the length-to-diameter ratio of the knot at its maximum in ated state. From the studies of nonequilibrium relaxation dynamics on prime knots cut at t = 0, we nd that even prime knots should be classied into dierent groups as (31; 51; 71; : : :); (41; 61; 81; : : :); (52; 72; 92; : : :), etc., based on their topological similarity

and their polynomial invariants such as Alexander polynomials. Our results suggest that the mathematical classication of knots can further be parametrized naturally into groups in a way that can have direct physical meaning in terms of structures and dynamics of knots. Furthermore, by scaling calculations, the nonequilibrium relaxation time is found to increase roughly as p12=5_.

This prediction is further supported by our data. 2000 Elsevier Science B.V. All rightsc reserved.

PACS: 61.41.+e; 83.10.Nn; 87.10.+e

Keywords: Polymer dynamics; Topological interactions; Knot classication

1. Introduction

The physical properties of many biological molecules such as DNAs, are strongly aected by their topological properties which play crucial roles in many

∗_{Corresponding author.}

E-mail address: [email protected] (P.-Y. Lai)

0378-4371/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved.

tum eld theory [8,9] in the last decades, the studies on the properties of physical knotted molecules are limited. There have been some studies concentrated on relat-ing the topological invariants of knots to the static properties of the knotted polymers such as developing phenomenological models [10] of the eects of knot complexity on the static and dynamic properties in terms of the number of essential crossings C. However, the ability of the theory to predict the equilibrium relaxation dynamics seems questionable. Grosberg et al. [11] presented a mean eld theory of the eect of knots on the equilibrium size of ring polymers. In their theory, a topological invariant was introduced which is related to the primitive path in the “polymer in the lattice of obstacles” model. These two works have consistent expression for the eects of topo-logical complexity on the variation of static chain conformations. They are also con-sistent with the experimental mobility test of DNA knots [12]. It was found that more complex molecules migrate faster than less complex ones which indicates that more complex molecules have more compact conformations. On the other hand, topological interaction manifests itself most prominently in dynamic phenomena in systems of en-tangled polymer coils or knotted ring polymers. The prohibition against chain crossing in a system restricts the number of its possible conformations. The only conformations are those that are topologically equivalent to one another as permitted by Reidemeister moves [6]. These possible conformations can only appear or disappear via continuous chain deformations without chain crossing nor breaking of the chain connection. Ring polymers which possess topological memory will strongly hinder their relaxation mo-tions. From these aspects, the systems of entangled polymer melts and knotted polymers are somewhat similar in their relaxation behavior.

In this study, not only the equilibrium structure of various types of polymer knots are investigated by Monte Carlo simulations, but also the nonequilibrium dynamic relaxation (untying) process of a knotted polymer cut at one point is studied. Experi-ments had shown [1,3] that a certain link in a ring DNA breaks up upon the action of topoisomerase and reconnects again after exchanging interlinked strands resulting in a knot structure. The relaxation dynamics of the knotted polymer is therefore important in such process. If the relaxation time is too fast, the knot will untie itself before

the link reconnects again and no knot structure will result. These results can be used to formulate theoretical models or to be compared with experiments. Our interest is focused on the system in which a ring polymer with a certain topological complexity is initially well equilibrated and then cut at a randomly picked link. The relaxation processes are monitored and analyzed.

2. Model and simulation details

Bead spring o-lattice model is employed in the simulation. N beads are connected by sti springs with interactions between the nonbonded beads through the square-well potential Unb=        ∞ (r ¡ ) ; − (6r ¡ ) ; 0 (6r) ; (1) where  and  are the energy and size parameters, respectively, and  = 1:5. The monomeric  and  are units used for the reduced quantities for temperature (T∗₌

kBT=) and distances. The interactions between bonded beads are represented by a

cut-o harmonic spring potential as Ub=1₂k2 r_ − 1:2

2

; 1:0 ¡_r61:4 (2)

and the potential is innite elsewhere. The parameters in the model are chosen to forbid any bond crossing to occur within the knotted chains. k2_{= = 400 and T}∗_{= 10 are}

chosen so that the system is in the good solvent regime. The system contains a single polymer chain with chain length N ranging from 42 to 82. We have studied the knotted polymers up to nine crossings: 31; 41; 51; 52; 61; 62; 63; 71; 72; 81; 91, and some composite

knots. The standard notation [6] for uniquely labeling a knot is CK where C is the

number of essential crossings and K is an index for a particular knot. Fig. 1 displays some of the knot types studied in this work. The simulations are performed under the conditions of constant temperature, volume and total number of beads. The initial congurations are generated by growing the chain bead by bead to the desired length and knot type. The trial moves employed for chains are bead displacement motions [13] which involve randomly picking a bead and displacing it to a new position in the vicinity of the old position. The distance away from the original position is chosen with probability that the condition of equal sampling of all points in the spherical shell surrounding the initial position must be satised. The new congurations resulting from this move are accepted according to the standard Metropolis acceptance criterion [14]. All runs are equilibrated for several million steps. Measurements for static properties such as radius of gyration are taken over a period of 1–4 millions MCS=monomer.

Fig. 1. Schematic knot diagrams. The (31; 51; : : :) group, the (41; 61; : : :) group and the (52; 72; : : :) group.

The Conway notation of each knot is also displayed. The rst integer in the Conway notation is the number of crossings in the braid structure while the second integer is the number of crossing in the right part of the knot diagram.

The mean radius of gyration

hRgi =

*

+

(xcm; ycm; zcm) are the coordinates of the center of mass and h i denotes the ensemble

average. The knotted ring polymer is allowed to equilibrate for a long time before it is cut randomly at one bond at t = 0. The nonequilibrium relaxation process is characterized by the time dependence of the radius of gyration Rg(t) as it approaches

its long time limit. Averages over dierent realizations (typically ∼ 300–500) of the relaxation processes are performed. Time is measured in units of Monte Carlo steps per monomer (MCS=monomer), one MCS=monomer means that on average every monomer has attempted to move once.

3. Knot complexity and equilibrium knot size

The number of essential crossings (C), i.e., the minimal number of crossings that the knot possesses no matter how one tries to untie it without cutting the string, is the simplest measure of the knot complexity. Quake [10] pictured a knot as a set of interlocked loops with the number of loops determined by the number of essential crossings, predicted the scaling law for the radius of gyration as Rg˙ NC1=3−. For

good solvents, Flory theory gives  = 3

5 and Rg ˙ N3=5C−4=15. Quake has performed

Monte Carlo simulations and the result is in agreement with the scaling law. How-ever, C is a fairly weak topological invariant. As we know there are 7 knots with 7 crossings and 166 knots with 10 crossings. The number of knots increases rapidly with the number of crossings. Grosberg et al. [11] recently introduced a new topological invariant p dened as the aspect ratio of the length(L) to the diameter(d) of a knotted polymer at its maximum in ated state, p = L

d. p has a greater value for more

com-plicated knots. It has been demonstrated [15] that p distinguishes rather well between dierent knot types and thus is a better topological invariant than C. For example, the knot 61 (p = 29:3) is less complex than 63 (p = 30:5) and the knot 81 (p = 37) is

more complex than 819 (p = 31). Flory approach has been applied [11] to estimate

the equilibrium polymer size by balancing the rubber-like elasticity and volume in-teractions between monomers and they found that there are four dierent regimes in which the polymer size has dierent dependence on p, N and T. The four regimes are the good solvent regime, the quasi-Gaussian regime, the poor solvent regime and the maximum tightened knot regime. In the good solvent regime, Rg∼ Np−4=15. We

have performed simulations to verify this relation. In Fig. 2a, hRgi versus p in a log–

log plot is shown for chains of various values of N. As a reference, hRgi=N' 0:38

for the trivial knot 01. Two groups of knots are observed from our data. One group

contains the prime knots (31; 41; 51; 52; 61; 62; 63; 71; 72; 81; 91) and the other group

con-sists of the composite knots; 31#31 (granny), 31#−31 (square), 31#41, and 31#51. Both

group show rather good linear relations with slopes relatively close to −4

15. However,

the values of the radius of gyration for the composite knots are systematically larger than the prime knots for N = 60 and 82. This result indicates that the static properties are not only determined entirely by the topological invariant p. Also from Fig. 2a and b, we can see that hRgi of 31#31#31(N = 82) deviates quite obviously from the linear

relation of the group (31#31; 31#41; 31#51). This further suggests that dierent number

of f actors in the composite knots, such as 31#31 and 31#31#31, may result in dierent

groups of composite knots. Also it has been recently found [16] that there exists exact additivity of the writhe number but subadditivity of p for the composite knots. They also showed that composite knots with two, three and four prime knots have dierent degrees of decits of p. Thus, it is plausible to assume that dierent groups of knots exist. In Fig. 2b, when the Rg is rescaled by N, we can see all the data collapse for

the prime knots as well as for the composite knots with two factors. This indicates that Rg ∼ N is a universal relation. However, it is noted that as p increases,

Fig. 2. (a) Radius of gyration (Rg) versus the aspect ratio (p) for knots at various chain lengths.

Dotted lines denote slope of −4=15. Filled symbols represent the prime knots (31; 41; 51; : : : 91):

() N = 42; (N) N = 60; (

•

•

31; 41; 51; 52; 62; 61; 63; 71; 72; 81; 91. (4) 31#−31; 31#31; 31#41; 31#51. ( ) 31#−31; 31#31; 31#41; 31#51;

31#31#31. (b) Same data as in (a) but plotted with Rg=N versus p.

over to the maximal tightened knot regime. In this regime, the polymer coils up so tightly almost as a compact ball and Rg becomes independent of p and solvent quality;

Rg∼ N1=3. The uniqueness of this regime is caused by the strong constraint imposed

on the knotted ring conformation.

4. Nonequilibrium relaxation and classiÿcation

Topologically distinct knots, even with the same value of C or complexity, dier from one another in the detail way of tying up the knot. Here we are interested in the question of what physical quantity can best re ect these dierences and manifests in some physically measurable quantities. As we have seen in the previous section that equilibrium structural quantities such as the size of the knot, could not resolve very well these dierences. Given a knot of a given number of essential crossings or complexity, if one wants to know how it was originally tied up, the simplest way is to cut the knot at some point and untie it to a linear string. Based on this simple idea, but on a molecular level, one can imagine untying the cut knot by Brownian motion and monitor the subsequent nonequilibrium relaxation dynamics. Here we shall focus only on prime knots. The knotted ring polymer is cut at a randomly picked link at t = 0 and the chain starts to relax towards the Flory coil conformation. Rg(t) denotes

Fig. 3. Rg(t) versus t for N = 60 for dierent knot types. Dotted lines are exponential ts. t is in units of

106 _{MC steps. The solid horizontal line is the ensemble average hRgi obtained from independent simulations}

of a linear chain of a chain of the same length at equilibrium.

of t is monitored. Fig. 3 shows the variation of Rg(t) versus t for N =60 with dierent

knot type. As we can see, these chains eventually reach their nal equilibrium state of the linear unknotted chain. The relaxation process can be well tted into an exponential behavior. The nonequilibrium relaxation time  is extracted from the data of Rg(t) by

assuming [Rg(∞) − Rg(t)]=[Rg(∞) − Rg(0)] decays as exp(−t=). It is worth noting

that this nonequilibrium relaxation diers considerably from the equilibrium correlation time of an uncut knot in which a long time mode exists for the time-autocorrelation function of nontrivial knots [17]. Our results on nonequilibrium relaxations do not show an obvious long time mode and no stretched exponential behavior is observed.

Fig. 4a shows the variation of nonequilibrium  versus p for various knots through dynamical Monte Carlo simulations.  for the trivial knot 01 is also shown for

com-parison. As we can see,  is highly nonmonotonic as p increases. Although the overall trend show, an increase in  with increasing p; the zig-zag-like local behavior is rather interesting. We believe that the local topological structure plays an even more important role in the relaxation dynamics of a cut knotted polymer as the degree of compactness does. In other words, knots with dierent detail topological structures relax in dier-ent ways. Remarkably,  shows a monotonic increasing behavior with p when these prime knots are divided into dierent groups, based on their topological similarity. These groups are (31; 51; 71; : : :); (41; 61; 81; : : :); (52; 72; 92; : : :), etc. (see also Fig. 1 for

these knot groups). Our data indicate the apparently puzzling fact that 31 has a longer

Fig. 4. (a) Monte Carlo data for the nonequilibrium relaxation time (in units of MCS=monomer=105_{) versus}

p for N = 60. (b) Same data as in (a) but  is separated into dierent groups. Solid curves are  ∼ p12=5_.

Table 1

The dierent knot groups with their Alexander polynomials
(s). C is the number of crossings in the knot. s is just an algebraic variable

Knot group Alexander polynomial
(s) C1: (31; 51; 71; : : :) (1 + sC)=(1 + s)

C1: (41; 61; 81; : : :) C₂ − 1 − (C − 1)s + (C₂ − 1)s2

C2: (52; 72; 92; : : :) C−1₂ − (C − 2)s +C−1₂ s2

value of p. This can be easily interpreted from our classication since 31 belongs to the

group of longer relaxation. As from Fig. 4b we can see that the group (31; 51; 71; : : :)

has longer relaxation times than other groups for knots with same chain length. This indicates that the (31; 51; 71; : : :) group has the strongest topological hindrance on the

relaxation moves among the groups studied. If we plot the relaxation time according to these groups, we nd smoothly increasing curves with  ∼ p12=5_{. An important}

out-come of these results is that the topological eect has a much stronger in uence on the nonequilibrium relaxation dynamics than on the equilibrium properties. The radius of gyration for the prime knots and even the equilibrium correlation time of an uncut knot [17], show a consistent monotonic behavior as a function of p while the nonequilibrium relaxation times need to be classied into dierent groups in order to have a regular monotonic dependence. Furthermore, by analyzing the Alexander polynomials (
(s)) of these groups of knots, we nd that the knots in each group have a similar form for their Alexander polynomials parametrized by the number of crossing C. These polynomials are listed in Table 1. One can easily see why the relaxation behavior are divided into groups, from the form of the polynomial invariants. The (31; 51; 71; : : :) group having

a long  can also be associated with the observation that the degree of
(s) increases with C, while for the other two groups their
(s)’s are always of degree 2.

The importance of the topological eect can be further veried from the outcome of a naive attempt to derive for the relation between  and p as follows. If the knot has a greater value of p (i.e., more compact), then simple calculations show that it has a higher free energy dierence from the Flory free coil state, and hence the relaxation process proceeds at a faster speed. But this result is contrary to what we have found in simulations. It is because of the existence of the free energy barrier that arose from the topological eect and is not accounted for in the naive free energy dierence approach. The system does not simply just relax downhill to the lower free energy Flory coil state, but has to overcome the barrier due to topological constraint of entanglement. Thus for the study of dynamical properties of knots, topological eect is a critical factor. When the chain is cut at t = 0, the closed ring constraint is relieved and the chain starts to relax. In good solvent regime, the chain would tend to expand out of its compact structure. However, the prohibition against chain crossing in a system hinders the process. Thus, the relaxation process can only proceed through some kind of reptation-like motion through its contour. However, it should be noted that the reptation move is somewhat dierent from the standard reptation theory. In the reptation theory, cross-linked network or polymer melts are considered and the monomers move in a “tube” resulted from the obstacles produced by other chains. The topology of the surrounding did not change signicantly in the intermediate time scales. From our simulation, we nd that the radius of gyration expands at the same rate as the end-to-end distance does. In other words, the diameter of the “tube” expands accordingly.

Employing the idea of a maximally in ated tube of contour length L and cross-section diameter d, the average time  taken by the chain to creep out of the initial contour length L can be evaluated on the basis of the reptation theory as  ∼ L2_{=D. The}

dif-fusion coecient D can be calculated according to the Einstein relation: D = kBT=t

where kB is the Boltzmann constant and t is the total friction coecient. As we

know, the friction coecient for reptation along a tube is proportional to the num-ber N of links in the macromolecule, i.e., N, where  is the monomer–solvent friction coecient in the Rouse model [18,19]. However, we believe that an inter-nal friction process is also involved as the chain varies its conformations during the relaxation process. For a linear polymer chain, the monomers tend to avoid each other in good solvents and the probability of two monomers in direct contact is small. How-ever, for knotted polymers, monomers are in close contact because of the existence of crossings. After cutting the knots, in the process of relaxation, monomers will slide onto each other, and extra friction will occur. The collision probability is greatly in-creased as the number of crossings increases. This monomer–monomer friction does not involve the solvent, but will be somewhat related to the viscosity of a uid of monomers. We use an analog to electric resistance to estimate this internal friction; that is the monomer-monomer friction coecient is assumed to be proportional to the ratio of length to cross-section area of the maximally in ated knot. Thus the total friction

Fig. 5. Nonequilibrium relaxation time  (in units of MCS=monomer) versus N for 41(N) and 51(

•

lines denote slopes of 1 + 2 ' 2:2.

coecient can be expressed as t=N+L, where  represents the monomer–monomer

friction coecient and  ' o=d2 for some characteristic monomer–monomer friction

o. Following the idea on the construction of the maximally in ated tube, Grosberg et

al. obtained L ∼ Rgp2=3 and d ∼ Rgp−1=3∼ Np−3=5. Then t= N + oN−p8=5 and

hence  ∼ L2_{=D ∼ N}2_p4=5_{[N + N}−_p8=5_

o]. For xed p and N1;  ∼ N1+2. For

xed N and p1;  ∼ p12=5_{. We have performed simulations and plotted  against N}

for 41 and 51 as shown in Fig. 5. We nd that the relation  ∼ N1+2 agrees

qualita-tively well with our simulations. Also, the relation ( ∼ p12=5_{) gives good description}

of our simulation data. However, in the long chain limit (N → ∞), one still recovers  ∼ N1+2_p4=5_{. The cross-over occurs at chain length of N ' p(}

o=)5=8. In the present

model in which an attractive square potential exists between monomers, physically we expect o  and the cross-over will occur and chain lengths are much greater than in

our present study.

5. Conclusions and outlook

The equilibrium structure of uncut knots and nonequilibrium relaxation of cut knotted polymers are studied by Monte Carlo simulations. Our results for the static quantities explicitly veried the scaling laws proposed by Grosberg et al. In the good solvent regimes, the averaged radius of gyration for both the prime and composite knots scale as Rg∼ Np−4=15where p is a topological invariant representing the length-to-diameter

ratio of a knotted polymer at its maximum in ated state. Although p is a better topolog-ical invariant than the crossing number C, our results for the composite knots indicate that p may not be the determinant invariant in correlating the static properties of all kinds of knots. Simulation results show that the radii of gyration of composite knots are always larger than the prime knots with roughly the same value of p. We also found that the dependence of the radius of gyration on p gradually becomes weaker as p increases. We believed that the knotted polymer chain is crossing over to the max-imally tightened regime. In the tight knot regime, the polymer is already in extremely compact situation so that an increase of p has no eect on its conformational change. For the studies of the nonequilibrium dynamics, we monitored the relaxation process of a fully equilibrated prime knotted polymer cut at a randomly picked link. When the constraint of the ring conformation is relieved, the chain starts to relax toward the Flory coil. Especially in good solvent condition, the driving force for the chain to expand is fairly strong. Yet, as we know the self-crossing of the macromolecular sections are forbidden and this eect greatly hinders the relaxation process. As a result, the relaxation proceeds through the reptation-like motions of the chain ends, especially for relatively compact knots. Perhaps the most important nding of this work is that the local topological structure plays a more important role in the relaxation dynamics of a cut knot than p does. The topological eect has much powerful in uence on the nonequilibrium relaxation dynamics than on the static or equilibrium properties of knots. The members of each group have similar patterns topologically as can be observed directly by careful visual inspection (see Fig. 1) or from their Alexander polynomials. The members in the (31; 51; 71; : : :) group have great resemblance in their

knot conformations and their Alexander polynomials can be parametrized by a single formula. Other groups like (41; 61; 81; : : :) and (52; 72; 92; : : :) can also be recognized

quite easily. The feature of long time mode is especially obvious for 31 as compared

to 41; 61; 81. Our results also showed that group (31; 51; 71; : : :) has a much longer

relaxation time than group (41; 61; 81; : : :) does. Somehow, in relaxing its conformation,

the group (31; 51; 71; : : :) has a topological hindrance that is inherently stronger than

other groups. The relaxation time is found to smoothly increase as  ∼ p12=5_{. Naive}

attempt of estimating the relaxation time from the free energy dierence of initial knotted state to the nal linear coil state failed indicating the presence of the free energy barrier due to topological eects. On the other hand, the idea of a maximally in ated tube turned out to be useful and analysis based on the reptation theory showed a promising outcome. The average time for a reptating chain to move along its tube by a length L is assumed to be  ∼ L2_{=D. In addition to the monomer–solvent friction}

force, the internal friction caused by monomer–monomer drag was also taken into account. The initial ring conformations and the existence of the essential crossings impose very strong constraint on the relaxation path of the chain. Therefore, for knotted chains, monomer–monomer contacts are inevitably frequent. The relaxation time is found to be proportional to N1+2 _{for xed p and N1, which agrees well with our}

simulation data. For xed N and p1;  ∼ p12=5 _{also gives a good description of the}

in it (not a closed chain), but the general physical behavior is not yet revealed and we are currently investigating this problem.

Acknowledgements

This research is supported by National Council of Science of Taiwan under Grant No. NSC 89-2118-M-008-003. Computing time provided by the Simulational Physics Lab., National Central University is gratefully acknowledged.

References

[1] W.R. Bauer, F.H.C. Crick, J.H. White, Sci. Am. 243 (1980) 118. [2] N.R. Cozzarelli, S.J. Spengler, A. Stasiak, Cell 42 (1985) 325. [3] S.A. Wasserman, N.R. Cozzarelli, Science 232 (1986) 951.

[4] Y. Arai, R. Yasuda, K.-I. Akashi, Y. Harada, H. Miyata, K. Kinoshita Jr., H. Itoh, Nature 399 (1999) 446.

[5] V.F.R. Jones, Bull. Am. Math. Soc. 12 (1985) 103.

[6] G. Burde, H. Zieschang, Knots, Walter de Gruyter, Berlin, 1985. [7] F.Y. Wu, Rev. Mod. Phys. 64 (1992) 1099.

[8] E. Witten, Commun. Math. Phys. 121 (1989) 351.

[9] L.H. Kauman, Knots and Physics, 2nd Edition, World Scientic, Singapore, 1993. [10] S.R. Quake, Phys. Rev. Lett. 73 (1994) 3317.

[11] A.Yu. Grosberg, A. Feigel, Y. Rabin, Phys. Rev. E 54 (1996) 6618.

[12] A. Stasiak, V. Katritch, J. Bednar, D. Michoud, J. Dubochet, Nature 384 (1996) 122. [13] Y.-J. Sheng, P.-Y. Lai, H.-K. Tsao, Phys. Rev. E 56 (1997) 1900.

[14] M.P. Allen, D.J. Tildesley, Computer Simulations of Liquids, Oxford University Press, New York, 1987. [15] V. Katritch, J. Bednar, D. Michoud, R.G. Scharein, J. Dubochet, A. Stasiak, Nature 384 (1996) 142. [16] V. Katritch, W.K. Olson, P. Pieranski, J. Dubochet, A. Stasiak, Nature 388 (1997) 148.

[17] P.Y. Lai, preprint (2000).

[18] P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. [19] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1992. [20] A.M. Saitta, P.D. Soper, E. Wasserman, M.L. Klein, Nature 399 (1999) 46.