bernard d. coleman and david swigon- theory of self-contact in kirchhoff rods with applications to...

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doi: 10.1098/rsta.2004.1393, 1281-12993622004Phil. Trans. R. Soc. Lond. A

Bernard D. Coleman and David Swigonsupercoiling of knotted and unknotted DNA plasmidscontact in Kirchhoff rods with applications toTheory of self

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10.1098/rsta.2004.1393

Theory of self-contact in Kirchhoff rods with

applications to supercoiling of knotted and

unknotted DNA plasmids

B y B e r n a r d D . C o l e m a n1

a n d D a v i d S w i g o n2

1Department of Mechanics and Materials Science and2Department of Chemistry and Chemical Biology, Rutgers University,

Piscataway, NJ 08854, USA ([email protected]; [email protected])

Published online 14 May 2004

There are circumstances under which it is useful to model a molecule of duplex DNAas a homogeneous, inextensible, intrinsically straight, impenetrable elastic rod of cir-cular cross-section obeying the theory of Kirchhoff. For such rods recent research has

yielded exact analytical solutions of Kirchhoffs equations of mechanical equilibriumwith the effects of impenetrability taken into account, and criteria have been derivedfor determining whether an equilibrium configuration is stable in the sense that itgives a strict local minimum to the elastic energy. This paper contains a summaryof published results on equilibrium configurations for the case in which a rod hasbeen pre-twisted and closed to form a knot-free ring. Emphasis is placed on the waythe writhe Wr of the ring, the number of its discrete points of self-contact, and thepresence or absence of lines of contact, depend on the excess link, Lk, which is ameasure of the amount the rod was twisted before its ends were joined. Bifurcationdiagrams are presented and a summary is given of the properties of the primary,secondary and tertiary branches that arise by successive bifurcations from the triv-ial branch comprised of configurations for which the axial curve is a circle. New

results are presented in the theory of equilibrium configurations of closed rods withthe topology of torus knots. It is remarked that examples of equilibrium configura-tions of closed rods of one knot type can be obtained from examples of other knottypes using methods previously employed to calculate isolas of equilibrium configu-rations of knot-free rings. Bifurcation diagrams are shown for supercoiled (2, 3) torusknots (trefoil knots). It is observed that for sufficiently large and sufficiently smallLk the minimum elastic energy configuration of a trefoil knot contains plectonemicloops with straight contact lines, although the configuration that minimizes the elas-tic energy of a general (2, q) torus knot over the entire range of Lk has self-contactalong a closed curve. As the ratio of the diameter of the rod to its length approacheszero, that contact curve becomes a circle, and there is an open interval of values ofLk for which stable equilibrium configurations with such circular contact curvesexist. Examples of minimum energy configurations are presented for both torus knotsand catenates formed by linking two unknots.

Keywords: DNA topology; elastic rods; contact problems

One contribution of 16 to a Theme The mechanics of DNA.

Phil. Trans. R. Soc. Lond. A (2004) 362, 12811299

1281

c 2004 The Royal Society

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1282 B. D. Coleman and D. Swigon

1. Introduction

A duplex DNA molecule contains two complementary polynucleotide strands joinedtogether in a WatsonCrick double-helical structure. Each of the strands has a sugarphosphate chain to which there are attached nucleotide bases of four types: A, T, C,G, with A complementary to T, and C complementary to G. The duplex structureforms when the bases on one strand bind to their complements on the other. Theresulting base pairs are, in an approximate sense, flat, rigid, rectangular objects,which are stacked with their mid-planes separated by 3.4 A and their centres ona curve called the duplex axis. In the form DNA assumes under conditions thatmimic those in living cells, each base pair is rotated relative to its predecessor byca. 34. In a rough sketch, the DNA duplex structure appears as a tube with anapproximate diameter of 20 A and with two parallel helical indentations, called themajor and minor grooves. The base pairs are in the interior of the tube, while thesugarphosphate chains lie on its surface and constitute the material between the twogrooves. The base pairs, or, equivalently, the bases on one of the two complementarystrands, are the units of the genome.

A DNA molecule that is closed, in the sense that each strand forms a closed curve,

is called a plasmid or DNA ring, and is said to be circularized. A bacterium has nonucleus and its entire genome is in a single plasmid. In a cell with a nucleus, theDNA is compacted into chromosomes in which it is anchored at several sites in sucha way that the segments between the sites are topologically equivalent to plasmids.

The attainment of an understanding of the way in which highly compacted DNA inbacteria or cell nuclei is made available for the processes of transcription, replicationand recombination is, in part, a problem in theoretical mechanics. This becomesclear once one notes that the configuration, and hence the compaction, of a plasmiddepends on a topological parameter, Lk, defined as the Gauss linking number of twoclosed curves: the duplex axis and (an arbitrarily chosen) one of the two strands thatform the DNA double helix. Plasmids that have the same size and base-pair sequencebut differ in such topological properties as the knot type of the duplex axis and the

linking number Lk are called topoisomers. The enzymes that convert one topoisomerinto another are called topoisomerases.

The various types of topoisomerases differ in their function (see, for example,Wang 2002). Topoisomerases of type I cut and ligate a single strand. Whereas topoi-somerases of type IA can increase Lk by 1, those of type IB can raise and lower Lkand bring a mixture of topoisomers into an equilibrium state in which the ratio ofthe concentrations of two topoisomers with different Lk and the same knot type isgiven, in accord with the laws of chemical equilibrium, by an exponential functionof their free energy difference. Topoisomerases of type II cut and ligate both strandsand change Lk by 2. The enzyme called DNA topoisomerase IV is of type II and canchange both linking number and knot type.

We here discuss an elastic rod model for DNA which is based on the assump-

tion that the rod under consideration obeys Kirchhoffs theory and is homogeneous,inextensible, straight when stress free, and transversely isotropic. In such a model,the rod axis C is identified with the duplex axis; the vectors d that are embeddedin the rods cross-sections and are employed to define twist density are identified

In crystals this rotation angle is closer to 36.

Phil. Trans. R. Soc. Lond. A (2004)



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Supercoiling of knotted and unknotted DNA plasmids 1283

with vectors that are normal to the duplex axis and point from that axis to thesugarphosphate chain of one of the two DNA strands.

In early attempts to employ elastic rod models to calculate the dependence of DNAconfigurations on Lk a difficulty was encountered. The configurations of principalinterest are supercoiled configurations in which the DNA makes contact with itself,

and, for even the simplest case of Kirchhoffs theory of elastic rods, the problem offinding with precision and analysing the stability of equilibrium configurations inwhich self-contact occurs was open.

After investigations by Le Bret (1984) and Julicher (1994) of configurations withself-contact in the theory of closed rods with zero cross-sectional diameter, Stumpet al. (1998) used an approximate method to calculate, for rods of non-zero diam-eter, configurations in which lines of contact are present. In our research on theseproblems, which has had as its goal the attainment of explicit and exact expressionsfor supercoiled configurations of rods of finite diameter, we have made use of knownanalytical expressions for the configurations of contact-free rod segments (see, forexample, Ilyukhin 1969; Landau & Lifshitz 1970) and a newly derived exact expres-sion for the configuration of a segment that is in contact with another along a straightline (Coleman & Swigon 2000). Also preliminary to our study of contact problems

was the development of a theory of the influence of end conditions on self-contact inDNA loops (Tobias et al. 1994; Coleman et al. 1995). That theory, as it permits oneto relate both the shape and elastic energy of a DNA loop to geometric boundaryconditions imposed on it by, say, a DNA-binding protein, has been applied to theproblem of calculating the configuration of small rings of DNA in mononucleosomesand has suggested a method of calculating DNA-histone binding energies from mea-sured equilibrium distributions of linking number (Swigon et al. 1998; Tobias et al.2000).

2. Variational inequalities for equilibriumconfigurations of impenetrable rods

As is the case for Signorinis problem in the linear theory of three-dimensional elasticbodies, the self-contact problem in nonlinear rod theory leads to a consideration ofvariational inequalities.

Here, as in recent studies (Tobias et al. 2000; Coleman et al. 2000; Coleman &Swigon 2000), we treat the self-contact problem for the case of a closed impenetra-ble rod of circular cross-section that obeys Kirchhoffs theory and is homogeneous,inextensible, intrinsically straight and transversely isotropic in such a way that it has

just two independent elastic constants: a bending modulus A and a twisting modulusC. The configuration of a rod R of that type is characterized by giving: (i) the axialcurve C, which is described by a smooth function x() with x(s) the spatial locationof the material point on C with arc-length parameter s, and (ii) the twist density ,which is, by definition,

= (s) = d(s) d

(s) t(s), 0 s 1. (2.1)Here l is the length ofC, t(s) = x(s) is the unit tangent vector for C at s, and d(s) isa unit vector imbedded in the cross-section ofR at s. As R is assumed to be closed,i.e. is a ring (which need not be knot-free), in each configuration

x(l) = x(0), t(l) = t(0), d(l) = d(0). (2.2)




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We write u for the twist density in a stress-free reference configuration. For anarbitrary configuration, the total twist Tw and the excess twist Tw (in turns) are

Tw = Tw(()) = 12

l0

(s) ds, Tw =1

2

l0

(s) ds, (2.3)

where = u is the density of excess twist. The total elastic energy ofR isthe sum of a bending energy B that depends on the curvature ofC and a twistingenergy T that depends on :

= B + T, B =12

A

l0

(s)2 ds, T =12

C

l0

(s)2 ds. (2.4)

We assume that R is impenetrable, that cross-sections ofR are circular and of uni-form diameter D, and that when self-contact occurs the contact forces are frictionlessreactive forces normal to the surface ofR.

A closed rod is subject to the topological constraint that all of its configurationsgive the same value to Lk, the Gauss linking number for two closed curves: C and thecurve

C obtained by displacing each point x(s) of

Calong d(s) by a fixed distance

less than 12D. There are several equivalent ways to define Lk; one way is to setLk equal to one-half of the total number of signed crossings of C and C seen in aprojection of the two curves on an (arbitrary) plane. It follows from a result of White(1969) and Calugareanu (1961) that the topological constant Lk obeys the relation

Lk = Wr + Tw, (2.5)

in which Wr is the writhe of the (closed) curve C and equals the average, over allorientations of a plane, of the sum of the signed self-crossings of the projection of Con the plane (Fuller 1971). Of course, although Lk is an integer, Wr and Tw neednot be.

Equivalent to equation (2.5) is the relation

Lk = Wr + Tw, (2.6)in which Lk, called the excess link, is, by definition, Lk Tw(u()) and is atopological constant that need not be an integer.

A pair (C, ) is called a configuration only if it obeys the constraints imposed onthe rod; these include the end conditions (2.2), the assumption of impenetrability,and the assumed knot type and value of Lk. A homotopy H : (C, ) ofconfigurations is said to be admissible if it is compatible with the constraints. Thefamiliar definition of an equilibrium configuration, in which a configuration (C, )is said to be in equilibrium if, for each smooth admissible homotopy H with a domaincontaining the point = 0 in its interior and with

(C, )|=0 = (C, ), (2.7)

there holds d

d(C, )

=0

= 0, (2.8)

is not appropriate when (C, ) is a configuration in which the rod makes contactwith itself. Such a configuration is in equilibrium if, for each admissible homotopy




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H obeying (2.7) with a domain of the form 0 , there holds the variationalinequality

d

d(C, )

=0

0. (2.9)

When (C, ) is in equilibrium according to this criterion, the equation (2.8) holdsfor those homotopies that, for small > 0, can be smoothly extended from 0 < to < < .

We are considering a rod R of length l that was closed to form a ring. If the ringis knot-free, the excess link Lk in equation (2.6) will be a natural measure of theamount that the rod was pre-twisted before it was closed to form a ring, and for eachvalue of Lk there will be an equilibrium configuration in which C is a circle andhence Wr = 0. For Lk sufficiently large, there will be equilibrium configurationswith Wr = 0 in which the ring makes contact with itself. Whether or not the ring isknotted, at a point of self-contact, say, that at which the cross-section with s = s

touches the cross-section with s = s = s, there holds|x(s) x(s)| = D, t(s) (x(s) x(s)) = 0. (2.10)

We shall here confine attention to cases in which a given cross-section is in contactwith at most one other. As we assume that the contact force, f (i.e. the forceexerted on the cross-section at s by the cross-section at s) is a reactive force thatis frictionless (and hence normal to the surface of the rod at s = s), we have

f = fx(s) x(s)

D. (2.11)

It can be shown that our present definition of equilibrium with as in (2.4) impliesthat throughout open intervals of values of s corresponding to contact-free sub-segments, there hold the equations

F = 0, M = F

t, (2.12)

in which M(s), the resultant of moments of the internal forces acting on a cross-section, is given by

M= At t + Ct, (2.13)and F(s), the resultant of the internal forces, is a reactive force not given by aconstitutive relation. In an early work on the subject, Kirchhoff (1876) observedthat equations (2.12) and (2.13) are mathematically equivalent to Eulers equationsfor the motion of a symmetric top, a fact which has been employed in research onDNA configurations (e.g. Benham 1977; Le Bret 1984). It is known that use of aparticular cylindrical coordinate system greatly simplifies the problem of obtainingan exact and explicit expression for a contact-free configuration of a rod segmentobeying (2.4) (e.g. Landau & Lifshitz 1970; Tobias et al. 1994).

Contact can occur at isolated points or along contact curves. If s

is an isolatedvalue or an endpoint of an interval Jof values ofs characterizing contact points, f,the contact force at s, is a concentrated force, and balance of forces and momentsyields

F(s + 0) F(s 0) + f = 0, M(s + 0) M(s 0) = 0. (2.14)




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In the interior ofJ the contact force has a continuous density f, equation (2.13)holds, and in place of the equations (2.12) one has

F(s) + f(s) = 0, M(s) = F(s) t(s), (2.15)with M and F again smooth functions of s.

In a contact-free subsegment ofR the equations (2.12) with M as in (2.13) are asystem of differential equations for C and with solutions that can be expressedin terms of elliptic functions (and integrals) and six parameters.

In cases in which R contains a pair of subsegments R, R that meet at a con-tact curve Cc, the relations (2.14) hold in the interior of the interval of values of scorresponding to the axial curve C ofR. If for one of these values of s we writev for the unit vector [x(s) x(s)]/D along the line connecting the centroids oftwo cross-sections in contact, write u for the unit tangent vector for Cc, and putw = u v, then in the equation

t(s) = u(s)cos (s) w(s)sin (s) (2.16)is the angle of winding of

C about

Cc. There are cases in which one can combine this

last equation with equations (2.10), (2.11), (2.13) and (2.15) to obtain a tractabledifferential equation for . When Cc is a straight line (i.e. a contact line), as is thecase for contact curves in knot-free rods, the vector u is independent of s, and thedifferential equation for , which then takes the form

=8

D2sin3 cos +

2C

ADcos2, (2.17)

has a solution,

(s) = arccot

q cot 0 p tan2(12 sn(s s0)

pq)cot 1

q p tan2(12

sn((s s0)pq))

(2.18)

in which p, q, 1 and the modulus of the elliptic function sn are functions of thematerial parameters D, C/A and the numbers 0, , where 0 is the value of ata point s0 where = 0. Thus, the configurations of the subsegments R and Rcan be expressed in terms of elliptic functions and four solution parameters, whichare 0, and the arc-length coordinates of the endpoints of C.

As R is a ring, ifR has n isolated points of self-contact and m contact lines, ithas 2(n + m) contact-free subsegments. Because there are six solution parametersper contact-free subsegment and four per contact line, the configuration of R isdetermined when 12n + 16m solution parameters are specified. The requirement thatLk have its preassigned value and the equations (2.2), (2.10), (2.11) and (2.14)yield 12n + 16m algebraic equations that can be solved for those solution parameters(Coleman et al. 2000). Thus one can obtain, for equilibrium configurations in whichself-contact occurs at isolated points and straight lines, (i) a value of (which the

governing equations require to be constant throughout R), (ii) a precise analyticalrepresentation for C, and (iii) the value off at each contact point.

See Coleman & Swigon (2000, pp. 188190) for a detailed derivation of (2.18), and Thompson etal. (2002) for a generalization of (2.17) to cases in which loads are imposed at the ends of the region ofcontinuous contact.




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In order for a solution of equations (2.10)(2.15), obtained as just described, tocorrespond to an equilibrium configuration it must be such that when the cross-sections at s and s are in contact, f in (2.11) is not negative; i.e. f, if not zero,tends to push apart cross-sections that contact each other. In addition, such a solutionmust obey the condition ofimpenetrability, i.e. must be such that if two distinct cross-

sections of the rod have a point in common, that point is on the boundaries of thetwo cross-sections. (Coleman & Swigon (2000, pp. 179, 180) give a mathematicallyequivalent formulation of this condition.)

Two configurations (C, ) and (C, ) are called equivalent if they have con-gruent axial curves and equal distributions of twist. Here, a configuration of a rodsubject to appropriate end conditions is said to be stable if it gives a strict local min-imum to in the class of configurations compatible with the imposed constraints.In other words, (C, ) is stable when, for an appropriate topology, it has a neigh-bourhood N such that (C, ) > (C, ) for each configuration (C, ) inN that is not equivalent to (C, ) and, in addition, is accessible from (C, ) byan admissible homotopy H.

A configuration (C, ) is differentially stable, if, for each admissible homotopy

Hwith domain 0 < and obeying (2.7), either

d

d(C, )

=0+

> 0 (2.19 a)

or

d

d(C, )

=0+

= 0 andd2

d2(C, )

=0+

0. (2.19 b)

(A differentially stable configuration obeys (2.9) and hence is in equilibrium.)Tobias et al. (2000) showed that (whether or not self-contact is present), when

(C, ) is a member of a family E of equilibrium configurations parametrized withLk, the following condition holds.

Condition (E). If (C, ) is stable, then, dLk/dWr, the slope at (C, ) ofthe graph of Lk versus Wr for the family E, is not negative.

This condition, although necessary, is not sufficient for even differential stability.However, in the same paper the following condition, condition (S), was shown to besufficient for stability as defined here.

Condition (S). A configuration (C, ) that is a member of a family E of equi-librium configurations is stable if

(1) on the graph of Lk versus Wr for the family E, dLk/dWr > 0 at (C, )and, in addition,

(2) (C, ) strictly minimizes bending energy in a locally uniform way, i.e. in thesense that (C, ) has a neighbourhood N such that, for each configuration(CE, E) in N and E, there holds B(C) > B(CE) for every accessibleconfiguration (C, ) in N for which C has the same writhe as, but is notcongruent to, CE.

A necessary condition for stability equivalent to condition (E) was formulated by Le Bret (1984).




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Verification of item 2 is a difficult matter. The following proposition (Tobias et al.2000) gives a necessary condition for stability, called condition (), that is strongerthan condition (E) and is sufficient for stability in several important cases.

Condition (). For each with 0 < l, let () be the minimum value ofdLk/dWr at (

C, ) over the families of equilibrium configurations of

Rthat

obey the added imposed constraint that the subsegment ofR with s < l be heldrigid. In order that an equilibrium configuration (C, ) be stable, it is necessarythat () 0 for each , 0 < l.

Once one has in hand an explicit analytical representation for an equilibrium con-figuration, one can calculate () for 0 < l and determine whether condition ()holds for that configuration. We have employed the theory of conjugate point criteriafor the stability of solutions of ordinary differential equations (e.g. Manning et al.1998) to prove that fulfilment of condition () is sufficient for the differential sta-bility of a contact-free configuration, and we are working on an extension of resultsof that type to configurations with self-contact. At the end of 3, in a discussion oftransitions between stable configurations, we shall assume that the fulfilment of the

slightly strengthened form of condition (), in which for each the relation () 0is replaced by the strict inequality () > 0, is sufficient not just for differentialstability, but also stability as defined here.

3. Configurations and bifurcation diagrams for knot-free rings

The results summarized above tell us that in the present theory it is useful to drawbifurcation diagrams as plots of Lk versus Wr. Some plots of this type, obtainedby the computational method discussed above and described in detail by Coleman &Swigon (2000), are shown in figures 1 and 2 for a knot-free ring with C/A = 7/5 and = 122, where = l/D. The chosen value of corresponds to a DNA molecule withD = 20 A that has 718 nucleotides in each of its two strands. The employed valueof C/A is compatible with measurements of the width of experimentally determinedequilibrium distributions of topoisomers of DNA plasmids (see Horowitz & Wang(1984) and the calculations of Frank-Kamenetskii et al. (1985)).

The reader may wish to compare the configurations and bifurcation diagramsshown here for a knot-free ring with those we presented in an earlier paper (Coleman& Swigon 2000) for such a ring with the same value of but with C/A = 2/3. Onp. 217 of that paper we derive implications of a rule (Tobias et al. 2000) relating theconfigurations and the bifurcation diagrams of rings with the same but distinctC/A. That rule, which is valid regardless of knot type or the presence or absence ofself-contact, states that there is a universal one-to-one correspondence between theequilibrium configurations of two rings with distinct values C/A that preserves theaxial curve

C(and hence the writhe Wr) and the value of (C/A)Tw .

Although calculation of Wr by numerical evaluation of its representation as a double integral overC is notoriously difficult, for the cases considered here one can obtain an analytical representation forthe integral along C of the geometric torsion and make use of the fact that Wr plus the torsion integralis an integer, called the self-link of C (Pohl 1968; see also Calugareanu 1961), which is not difficult toevaluate. For details see Swigon et al. (1998).




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A0

Wr

0

0

1

2

3

4

5

B0

C0

D0

E0

F0

A2

A1

A3A4

1 2 3 4 5

Lk

Figure 1. Graph of excess link, Lk, versus writhe, Wr , for the primary branch . Here and infigure 2, = 122. For n = 0, 1, 2, 3, the configurations with n points of self-contact correspond to

points between An

and An+1

. At values of Lk greater than that at A4

, the configurations havean interval and two isolated points of self-contact (which are indiscernible at this scale of draw-ing). Two examples are shown of configurations with Lk > Lk(A4), one with Lk = 3.55and the other with Lk = 5.38. Families of stable configurations are shown as heavy curves.Configuration A1 has been called the figure-of-eight configuration; those with Wr > Wr(A1)are said to be plectonemically supercoiled. The parameter Lk obeys equation (3.1) at config-urations A0, B0, . . . , of branch . Here and in figures 25 selected configurations are shown astubes of diameter 20 A. Here the line of view is at an angle of 45 to the axis of D2 symmetry.

For knot-free rings of the type here under consideration, the bifurcation diagramhas a branch (called the trivial branch and labelled ) that is comprised of configu-rations for which C is a circle and hence Wr = 0. Each point of with

Lk =A

C

m2

1, m = 2, 3, . . . , (3.1)is a bifurcation point at which a primary branch with Wr = 0 originates. The primarybranches with m = 2, 3, etc., are referred to as branch , branch , etc. For eachm 2 the symmetry group of the configurations in the primary branch with index mis the dihedral group Dm (which has order 2m). Hence, whether or not self-contact




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S

R

Q

P

Wr

0

1

2

3

4

5

6

B0

B1

A0

C0

D0

G0

H0

F0

E0

Lk

0 1 2 3 4 5 6

B2I

B2III

B2II

B2IV

B2I,II

B3

I II

II,I

III

IV

Figure 2. Graphs of Lk versus Wr for the primary branch (solid curve), the secondarybranches I, II , III , IV (long dashes) originating at , and the tertiary branch II,I (dots)originating at II. The points B

0, B1, P, Q, R are bifurcation points for branch , S is abifurcation point for branch II. For n = 0, 1, 2, 3, the configurations on branch with 3n pointsof self-contact (i.e. with one such point in each lobe) correspond to points between Bn and Bn+1.Not visible at this scale is a closed secondary branch V that meets at the bifurcation pointsQ and R. A detailed graph of such a closed branch and views of configurations on it are givenby Coleman & Swigon (2002). Here, the line of view for Bn (n = 1, 2, 3) is parallel to the axisof highest symmetry; for all other configurations it is perpendicular to that axis.

is present, the curve C for a configuration on the primary branch of index m hasa single m-fold symmetry axis that is perpendicular to a plane containing the 2mpoints at which the curvature of

Chas a local extremum. Each of the m lines that

intersect the m-fold symmetry axis and pass through two extrema of is a twofoldsymmetry axis. Each primary branch, with the exception of branch of figure 1,contains points of secondary bifurcation. Configurations on the secondary branchesoriginating at points on branch of figure 2, have C2 symmetry. Configurationson the tertiary branch II,I, previously called II (Coleman et al. 2000; Coleman &




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Swigon 2000, 2002), that originates at the point S of secondary branch II have nodiscernible symmetry.

Elsewhere (Coleman & Swigon 2000), we have described configurations on branch (for which m = 4) and some secondary branches that originate on that primarybranch.

The configurations in the interior of regions of branches that are drawn as heavycurves in figure 1 are stable. As condition (E) does not hold on the intervals of branch that are there drawn as light curves, the configurations in those intervals are notstable.

In figure 2 one sees the primary branch , the secondary branches that originateat points on , and the tertiary branch II,I. On those branches, condition () holdsonly for the configurations in the heavily drawn interval of branch that runs fromthe bifurcation point B1 to the bifurcation point P. No other configurations in thosebranches can be stable.

The bifurcation diagrams presented by Swigon (1999), Coleman et al. (2000) andColeman & Swigon (2000, 2002) have sufficient precision to reveal such features ofself-contact as the following: when changes in Lk result in first one and then twoisolated points of self-contact in a lobe, further change results in three isolated pointsand then a mixture of intervals and isolated points of self-contact.

A knot-free ring can have families of equilibrium configurations that are not onprimary, secondary or higher-order branches, i.e. that are isolas in the sense that theyare not connected to the trivial branch by a continuous path of equilibrium configu-rations that obey the constraint arising from impenetrability. The isolas studied byColeman & Swigon (2000) are connected to one of the secondary branches originatingon branch , or to the tertiary branch II,I, by one-parameter families of solutionsof equations (2.10)(2.15) that contain subfamilies that do not obey the requirementthat, if distinct cross-sections have a point in common, that point must lie on thesurface ofR. Such families occasionally contain subfamilies of configurations thatobey the constraint of impenetrability, but are knotted, and hence are not configura-tions of the original knot-free ring. (The heavy solid curve seen in fig. 24 of Coleman

& Swigon (2000) corresponds to a one-parameter family of trefoil knots obtained inthis way from the branch I of a knot-free ring.)

4. Configurations and bifurcation diagrams for torus knots

We now turn to the theory of equilibrium configurations of closed rods that have axialcurves with the topology of (p,q) torus knots, i.e. knots that can be drawn on thesurface of a torus as not self-intersecting curves that pass q times through the holein the torus and go p times around that hole, where p and q do not have a commondivisor and obey the relations q > p 2. The bifurcation diagrams we present in

For closed rods obeying the present assumptions, Domokos (1995) proved that each contact-freeequilibrium configuration has Dm symmetry, which is what we found. In the same paper he conjectured

that the symmetry group of an equilibrium configuration with self-contact contains C2 as a subgroup,which is not the case for configurations on the tertiary branch II,I originating at the point S.

Because of their importance in molecular biology, we should mention here that there are classesof catenated rings that can be treated as generalized torus knots for which p and q do have a commondivisor. When q is an even number, the generalized (2, q) torus knot is a link of two unknots, which wemay call the (2, q) torus link.




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this section are for closed rods with C/A = 7/5 and = 170 (which corresponds toa DNA molecule with D = 20 A and 1000 nucleotides in each strand).

Langer & Singer (1984) gave convincing arguments to the effect that a closedrod can have a contact-free equilibrium configuration only if it is knot-free or hasthe topology of a torus knot. Although they present their arguments assuming that

the elastic energy is the integral of 2

(they call rods with that property elasticcurves), their proof rests on the validity of a second-order differential equation for(s) which has been shown (Coleman et al. 1993) to hold for contact-free equilibriumconfigurations (and also travelling waves) of rods of the type that we consider here(i.e. rods for which the twisting modulus C in equation (2.4) need not be zero). Inother words, the propositions of Langer & Singer about the stability of contact-freeconfigurations have a broader range of validity than originally claimed. Contact-freeconfigurations of torus knots were calculated by Le Bret (1984) and Starostin (1996).

The relation between the families of contact-free equilibrium configurations of torusknots and contact-free regions of primary branches of the bifurcation diagram for aknot-free ring is shown in figure 3, in which one sees the trivial branch , segmentsof the branches , , , , (for which m in equation (3.1) is, respectively, 2, 3,

4, 5, 6), and families of contact-free equilibrium configurations of (m 1, m) torusknots with m = 3, 4, 5 that are labelled , , . Each family of (m 1, m) torusknots is connected to the primary branch for a knot-free ring with the same m bya one-parameter family of solutions of the equations (2.10)(2.15) do not obey theimpenetrability constraint. Each such family contains a point (labelled / in figure 3)at which the (axial) curve obtained by solving (2.10)(2.15) passes through itself andhence Lk and Wr undergo a jump equal to the integer m2 m.

A contact-free equilibrium configuration of a rod with the topology of a (p,q) torusknot has Dq symmetry. The rod can attain a contact-free configuration if and onlyif the value of is larger than a critical value that depends on p and q. We findthat (2, 3) = 26.47, (3, 4) = 63.33, (4, 5) = 120.19, (5, 6) = 198.60. As thevalue of employed for the calculations shown in figure 3 is 170, there a (5 , 6) torusknot can have no contact-free configurations.

Langer & Singer (1984) and Le Bret (1984) conjectured that contact-free configu-rations of torus knots are unstable. As evidence tending to support this conjecture,we observe that the families of contact-free equilibrium configurations of (m 1, m)torus knots shown in figure 3 do not obey condition (E) and hence contain onlyunstable configurations.

We now focus on equilibrium configurations of trefoil (i.e. (2, 3) torus) knots whichoccur in significant concentration in DNA cyclization experiments under a broadrange of conditions. As the full bifurcation diagram for trefoil knots has a complextopology with multiple bifurcation branches forming closed loops, a complete descrip-tion of it is not presently available. Following the dissertation of Swigon (1999), wehere confine attention to the set M of equilibrium configurations defined as follows.

A tilde, as in B1 and , generally indicates that the configuration (or family of configurations)under consideration is knotted (or is comprised of knotted configurations). An exceptional case is thatof the family , which is topologically equivalent to the branch of a knot-free ring.

Of course, > , where is the smallest value of for which a configuration without penetrationexists in a class of closed rods of prescribed knot type. Configurations with = are called ideal knotsby Katritch et al. (1996) and maximally tightened knots by Grosberg et al. (1996).




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A0

C0

D0

E0

B0

0

1

2

3

4

5

Wr+ J

Lk+J

0 1 2 3 4 5 6

E1

D1

D0

D1

~

C0~

C1~

B1~

B0~

A0~

A1~

C1

B1

~

A1//

//

/

Figure 3. Graphs of Lk+J versus Wr +J for one-parameter families of contact-free equilibriumconfigurations of a closed rod. When the rod is knot-free, J = 0; when it has the topology of a(m 1, m) torus knot, J = m2 m. Here and in figure 4, = 170. The families of knot-freeconfigurations, , , , (solid curves) are connected to the families of torus knots: , , (solidcurves) by families of configurations that fail the condition of impenetrability (dotted curves).

Configurations in the family have the topology of (2, 3) torus knots (i.e. of trefoil knots), in of (3, 4) torus knots, and in of (4, 5) torus knots. Although the configurations in are mirrorimages of configurations in and hence are not knotted, J = 2 for . With the exception of ,all the families shown contain only unstable configurations. As the passage of the axial curvethrough itself (at the point marked with /) results in a jump in Lk and Wr by the amountmm2, each graph of Lk+ J versus Wr + J is smooth. Points corresponding to configurationsin which contact occurs without penetration are shown as hollow circles. Solid circles give thelocation of the depicted axial curves, and the line of view of is parallel to the axis of highestsymmetry. The axial curves for the torus knots are not all drawn on the same scale.

For each Wr an equilibrium configuration (C, ) is in M if and only ifC mini-mizes B(C) over the set of axial curves C with the same writhe (and of course thesame length and topology) as

C. The importance of

Mstems from the fact that it

contains every configuration that (globally) minimizes the total elastic energy forpreassigned Lk. However, M is strictly larger than the set of such global minimizersof .

One-parameter families of configurations in M are shown in figure 4. The familiescontaining configurations with only isolated points of self-contact form two discon-




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a

cb

a

a

a

a

b

c

a

a

bc

a

Wr

5 4 3 2 1

7

6

5

4

3

2

1

L

B0

O

M1

N1

K

C

Lk

~

N2~

N3~

N4~

~

II

I

I

III

C

~~

~

~

~

M2~

M3~

M4~

~~

B1

Figure 4. Graphs of Lk versus Wr for configurations in the set M of configurations of thetrefoil knot. (M is defined in the text.) One-parameter families of stable configurations areshown as heavy curves. The point B

0of the family (which is also shown in figure 3) is the

bifurcation point at which the families I and II originate. The family III is not connectedto by a continuous family of equilibrium configurations of the trefoil knot. The point B

1lies

on but not I. The configurations corresponding to points in the family C (heavy dashedvertical line) are stable; each has a contact curve that is close to a circle. Here, the line of viewfor B

0, B

1, K, C is parallel to the axis of highest symmetry; for all other configurations it is

perpendicular to that axis.

nected sets: a set M1 containing configurations with Wr < 3, and a set M2containing those with Wr > 3. The families in the set M1, labelled I and II, areconnected to the family of contact-free configurations (shown in figure 4 as a lightdotted curve and as a solid curve in figure 3) at the bifurcation points B

1and B

0,

respectively; both of those points correspond to configurations with three points of




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self-contact. (A more complete bifurcation diagram makes clear the fact that bothB1

and B0

are termination points for families of configurations with 1, 2, or 3 pointsof self-contact, but only I and II belong to M1 and are shown in figure 4.) Theconfigurations in the family I have two points of self-contact and C2 symmetry,while those in the family II have three points of self-contact and D3 symmetry.

Let us follow the familyI in the direction of decreasing Wr . The D3 symmetry ofthe configuration at B

0is broken once one moves away from B

0and the configurations

in I between B0

and K have two points of self-contact and C2 symmetry. Thesymmetry is broken again at the point K, with the configurations between K andL showing two contact points but no discernible symmetry, and regained at thepoint L, with the configurations between L and M1 showing three contact pointsand C2 symmetry. In figure 4 we have given the labels a, b and c to the threepoints of maximum curvature (and to the loops containing those points) to help thereader to visualize the transition B

0 K L. For the configuration at K, the axisof symmetry is perpendicular to the plane of the paper and passes through b, whileat L that axis lies in the plane of the paper and passes through a. At M

1a new point

of self-contact appears in the loop labelled a, and the configurations between M1

and M4

follow the rule for the formation of points of self-contact in a plectonemicloop: the one point of self-contact in loop a splits into two points at M2, at M3 a thirdcontact point appears between the two, and for configurations with Wr < Wr(M

4)

the middle contact point spreads into a straight line of contact.The situation is similar for the family III , the one family in the set M2. The

configuration N1

in III resembles the configuration M1

in I, except for a differentsign of crossing at the self-contact point in the loop labelled a. As one traversesthe family III in the direction of increasing Wr, one again finds that configurationsfollow the rule for the formation of points of self-contact in a plectonemic loop, withthe configurations between Nn and Nn+1 showing n self-contact points in the loopa (i.e. n + 3 total points of self-contact). Computational complexities prohibited usfrom obtaining configurations with 3 < Wr < Wr(O).

The writhe of configurations in the family BII originating at the point B0

increases

by a nearly undetectable amount as one traverses that family away from B0

in thedirection of increasing Lk. These configurations all have Wr < 3 and are in theset M. Although the shape of the axial curve changes very little, new points of self-contact do appear. The configuration C, which contains 12 points of self-contact, isat the limit of feasibility of our method of computing the integration constants inour analytical solution of the equations (2.10)(2.15). We do not know how largethe value of Lk is at which the family II ceases to be a subset ofM1, i.e. ceasesto be made up of configurations with only isolated points of self-contact, but we doknow that there is an interval of values of Lk (each greater than Lk(C)) for whichthe minimum energy configuration of a trefoil knot is one showing self-contact alonga closed curve. That one-parameter family C of configurations in M with closedcontact curves is shown in figure 4 as a heavy dashed line.

As exact analytical expressions for equilibrium configurations of rods with self-contact along general curves are not available, we now derive approximate expressionsthat are appropriate to rods with large , i.e. small d = 1 = D/l.

Let us consider a ply Pformed by two rod segments R, R of equal lengths thatmeet at a contact curve Cc. When the two cross-sections at one end ofPare glued tothe two cross-sections at the other end, Cc becomes a closed curve, and the segments




17/20


R, R form either (i) one closed rod R (with each end ofR glued to an end ofR) or (ii) two closed rods R, R (with the ends ofR glued together, and theends ofR glued together). We here consider the cases in which Cc is not knottedand hence, in case (i), R is either knot-free or a (2, q) torus knot, and, in case (ii),R and R are knot-free and either unlinked or form a (2, q) torus link in which theaxial curves ofR

and R

can be drawn as nonintersecting curves on the surface ofa torus with each passing once around the hole in the torus and q/2 times throughthat hole.

The configuration ofP is characterized by the contact curve Cc (with arc-lengthcoordinate sc) and an angle = (sc) defined as follows. As in 2, we write v(sc) forthe unit vector along the line connecting the centroids of two cross-sections in contactat sc, and u(sc) for the unit tangent vector for Cc at sc, and we put w = uv. Thecurvature c and torsion c ofCc and the angle obey the relations

c sin = w u, c cos = u v, c + = v w. (4.1)

It can be shown that when the diameter D ofR and R is much smaller than theirlength l, i.e. when d 1, the total elastic energy ofPobeys the relation

= Al

c

0

2c dsc + 42C

lc( (Wr)c)2 + O(d2), (4.2)

where lc and (Wr)c are, respectively, the length and the writhe of Cc, and = 1

2(Lk q), (4.3)

with Lk the excess link ofR (and hence a topological constant independent ofthe deformation of P). In (4.3) the minus (plus) sign is chosen for right-handed(left-handed) (2, q) torus knots and links.

When we compare equation (4.2) with equation (2.4) and make use of the fact thatfor a closed rod in equilibrium is a constant equal to = 2(Lk Wr)/l,we conclude that (4.2) is identical to the equation for the total elastic energy ofa hypothetical knot-free closed rod with length lc, bending modulus 2A, twistingmodulus 2C and linking number that is in equilibrium with curvature c = c(sc),writhe (Wr)c and (spatially uniform) twist density c = 2( (Wr)c)/lc. As itis known that, for || < (A/C)3, the minimum energy configuration of such a rodis a circle, one may conclude the following.

Proposition 4.1. If the minimum energy configuration of a (2, q) torus knot orlink with (A/C)23 q < Lk < (A/C)23 q shows self-contact along a closedcurveCc, then, in the limit as d 0, Cc is a circle.

When Cc is a circle, the balance equations (2.15) yield, in the limit as d 0, afourth-order nonlinear differential equation for (sc),

6()2

3cC

A

+ 12

2c(cos 2

3) + 122c(()2

2c)sin2 = 0, (4.4)

with the end-condition(lc) (0) = q; (4.5)

here the plus sign is chosen for right-handed (2, q) torus knots and links. In orderfor a solution of (4.4) and (4.5) to correspond to a stable equilibrium configuration




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(2,3) (2,4)

(2,5) (2,6)

(2,7) (2,8)

Figure 5. Configurations of (2, q) torus knots and linksthat minimize over the entire range of Lk.

ofP, f in (2.11) must not be negative. The balance equations (2.15) imply that, inthe limit as d 0,

f = AD2

+

2c2

(sin 2) ()4 + 2c

2(cos2 + 3)()2

4c

2(1 + cos 2)

CcD2

[()3 2c(1 + cos 2)]. (4.6)

The equations (4.4) and (4.5) can be solved by expansion of the inverse functionsc = sc() as a sum of a term linear in and terms linear in sin 2n for n = 1, 2, . . . .For the left-handed trefoil knot, which has q = 3, we find that

sc = 23c

( + B1 sin2 + B2 sin4 + ), (4.7)




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Table 1. The coefficients B1 and B2 in the solution (4.7) ofequation (4.4) obeyed by trefoil knots with d small

(Here q = 3, C/A = 7/5, 1/d = 170.)

Lk B1 B2 f

min

1.64 7.28 103 2.25 105 03.00 1.32 102 6.72 105 0.40AD/c3.97 3.03 102 3.26 104 0

with B1, B2, . . . constants that depend on Lk and C/A. Values of B1 and B2 forselected values of Lk are given in table 1. By calculating fmin, the minimum value off in equation (4.6), we find that for the trefoil with the present values ofC/A and the solution (4.7) describes a stable equilibrium configuration only if3.97 < Lk

bernard d. coleman and david swigon- theory of self-contact in kirchhoff rods with applications to...

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