Toroidal Elastic Supercoiling of DNA

C. R. CALLADINE, Department of Engineering, University of Cambridge, Cambridge CB2 IPZ, England

Synopsis

Covalently closed circular DNA can exist in different configurations known as circular, toroidal, and interwound. Changes among these forms can be made in several ways, including the insertion of dye molecules between adjacent base pairs, which tends to untwist the dou- ble-helical structure. The aim of this paper is to discuss these configurations, and the changes among them, in the context of classical elastomechanics. The concepts of twisting, linkage, and writhing are explained. Simple experiments on a twisted linear-elastic rod are described, and it is shown that although the circular and interwound forms may be modeled in this way, the toroidal form does not occur, being mechanically unstable. Theoretical energy calculations by Levitt on bent and twisted DNA show that DNA exhibits a particular kind of nonlinear elasticity in which there is an unusual coupling between bending and twisting. The aim of the paper is to show qualitatively that this special kind of elasticity can stabilize the toroidal form of closed circular DNA.

INTRODUCTION

Covalently closed circular DNA molecules in aqueous solution can exist in a variety of geometrical config~rations.l-~ Physicochemical methods and electron microscopy suggest that the range of stable configurations includes the three classes (a) circular, (b) toroidal, and (c) interwound, which are sketched in Fig. 1.

Reversible transformations within and between these classes may be wrought in several ways, including changes in the pH of the environment and the intercalation of dye.2 In the latter process, the insertion of a dye molecule between adjacent base pairs in the double-helical structure of DNA locally alters the angle of twist, tending to unwind a length of linear

(b) Fig. 1. Three distinct configurations (not to same scale) thought to be adopted by circular

DNA: (a) circular, (b) toroidal, and (c) interwound.

Biopolymers, Vol. 19, 1705-1713 (1980) 0 1980 John Wiley & Sons, Inc. 0006-3525/80/0019- 1705$01 .OO

1706 CALLADINE

DNA with free ends; but in circular DNA such unwinding is frustrated, and in consequence there is a tendency for the configuration to transform from (a) to (b) or (c).

In order to understand the process of transformation from one configu- ration to another, it is necessary to describe not only the geometrical aspects of the process, but also the mechanical features. Benham5 has analyzed the toroidal form of circular DNA in terms of a model involving the dis- tortion of an initially straight isotropic rod. However, as Le Bret6 has pointed out, the toroidal configuration does not occur as a stable form for a linear-elastic rod; and simple experiments on straight rubber rods of square cross section tend to confirm this.

The aim of this paper is to propose an explanation for the stability of the toroidal form of DNA, in terms of the classical mechanics of nonlinear and linear-elastic systems.

The term “supercoil” is normally used to describe a configuration such as Fig. l(b), since straight DNA has a coiled (double) helical structure. In this paper I take DNA to be a plain rod and use the term coiled to describe what is usually called a supercoiled structure.

For the sake of simplicity I shall describe the changes of configuration in terms of the absorption of dye by a circle of DNA whose basic, dye-free, state is that of Fig. l(a). This is a rather artificial base state in terms of experimental work and is here adopted for the sake of convenience in presenting the argument. I shall also use the conventional expression ALk as a measure of the absorption of the dye; a L k is equal to the number of turns of unwinding which the dye would produce in a straight piece of DNA with free ends (see below).

GEOMETRY OF TWISTING AND COILING

It is convenient to introduce the general geometrical and topological aspects of twisting and coiling7-10 in relation to a particular transition from a circular to a toroidal configuration as follows. Consider a simply-closed ribbon, shown in Fig. 2(a), which has a straight section PQ that is destined to form a single coil of a toroid. Suppose that the ribbon is now held at P, cut at Q, given a complete turn of twist, and rejoined as in Fig. 2(b) with the section PQ in a state of uniform twist. The same strip is shown again in Fig. 2(c), but the point Q has now been moved relative to P along the path sketched: this maneuver is easy to demonstrate with a short strip of paper. There is practically no twist in the ribbon, which now has the form of a figure eight. Clearly, the loss of twist in section PQ has been compensated somehow by an increase in curvature. The ribbon is said to have writhed into a twist-free configuration. The concepts of writhing and twisting in a closed loop can be quantified precisely, and it may be shown in generalsJO that a writhing number W r and a twisting number T w are related to a third number L k by the relation

W r + T w = L k (1)

ELASTIC SUPERCOILING OF DNA 1707

(b) 0 n curvature k

Fig. 2. Geometry of twisting/writhing in a closed ribbon: (a) simple closed loop; (b) one turn of twist is introduced; (c) tranformation from (b) without tearing; (d) Cartesian plot of twist (T) against curvature ( K ) for the process b-c.

Lk, the linking number, is the number of times one edge of the ribbon is linked to the other edge, the two edges being regarded as separate curves for this purpose. In Fig. 2(a), Lk = 0, but in both (b) and (c) Lk = 1: Eq. (1) expresses the fact that Lk is a topological invariant of the process. It may seem paradoxical to use the conventional symbol ALk as described above for what is in fact a change of twisting number on account of the uptake of dye, since Lk actually remains constant in a covalently closed loop. However, we can see by Eq. (1) that a change of twisting number on account of the dye may be regarded as being precisely equivalent to a change of linkage number in a dye-free closed loop; and it is much easier to envisage physically-and investigate experimentally by means of elastic rods-an equivalent artificial process in which a dye-free loop is cut and a change of linkage inserted.

For present purposes it is most useful to consider mainly the curvature K and twist 7 of the section PQ as the transformation from Fig. 2(b) to (c) takes place through an assumed sequence of uniform helical configurations. The K , 7 path [Fig. 2(d)] may readily be shown to be a circle centered on the origin, with points B, C corresponding to configurations b, c, respectively, lying on the two axes. Both K and 7 are defined as the rate of change of an angle with respect to arc length s; in terms of a small “vehicle” moving along the ribbon, twist is d(ang1e of roll)/ds, while curvature is d(ang1e of pitch)/ ds. Here, roll and pitch are defined as rotations about forelaft and transverse axes in the vehicle, respectively. Thus, in traveling from P to Q in Fig. 2(b), the vehicle rolls through 28, whereas in Fig. 2(c) it pitches through 28; consequently, points B and C in Fig. 2(d) are equidistant from the origin. For a full study it is of course necessary to prescribe a sign convention for each of these variables; but this is not required for present purposes.

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ELASTICITY OF AN ORDINARY ROD

Consider a uniform linear-elastic rod whose principal bending stiffnesses are equal, e.g., a rod of circular or square cross section.11J2 When a short portion of such a rod is subjected to pure torque T and pure bending mo- ment M , it undergoes changes in twist and curvature according to the relations

where E I and GJ are the flexural and torsional rigidities of the rod, re- spectively. The mechanical strain energy U per unit length is given by

U = ~ E I K ~ + $GJr2 (3)

and thus contours of constant U in K , r space are concentric similar ellipses, as shown in Fig. 3(a).

Suppose now that the edges of the “ribbon” in Fig. 2 are merely lines inscribed on the surface of such a rod, and that at points P and Q external clamps are applied to the rod which completely prevent rotation but do not inhibit transition. In general, according to classical m e c h a n i c ~ , l ~ ~ 3 a configuration of static equilibrium is possible if U is stationary with respect to relative displacement of P and Q; and for the equilibrium to be stable, U must be a minimum. It follows immediately from a comparison of Figs. 2(d) and 3(a) that if we neglect the restraining influence of the rod outside the segment PQ, the only possible equilibrum configurations are those of Fig. 2(b) and (c) and that if E I < GJ, (c) is stable, whereas if EI > GJ, (b) is stable; and in the borderline case E I = GJ, both (b) and (c) and indeed all intermediate configurations are in a state of neutral equilibrium. In particular, we do not find a tendency for the rod to adopt a helical config- uration (apart from the limiting case of a flat or circular form).

Fig. 3. Contours of constant strain energy U in curvature, twist space. (a) Linear-elastic rod: the curves are ellipses [Eq. (3)]. (b) Form of U contour for DNA; sketch not to scale. Angle a marks the direction for which a radius intersects the curve orthogonally: it is the angle of the resulting helical coil, measured from a line parallel to the axis of the coil. (c) Contours for several levels of U with the geometric constraint of Fig. 3(d) superposed. The point marks the (stable) configuration for which U is minimum.

ELASTIC SUPERCOILING OF DNA 1709

Stability of a Straight Elastic Rod Bent Into a Circular Hoop

When we take a straight, linear-elastic rod, of, (say), square cross section, twist it uniformly by a given amount U k , and then bend it and join it into a circular hoop, we find that its behavior depends strongly on the value of aLk. A silicone rubber rod about 10 mm square and 300 mm long is con- venient for experiments of this kind: self-weight effects are not important and a square cross section makes it easy to count U k . (Bunsen-burner tubing is not suitable, since it is curved in its relaxed state.)

In experiments of this sort we find that for A L k = 0 and 1, the hoop is stable against both small and large perturbations. When a L k = 2, it is stable against small perturbations, but when a sufficiently large twist is imposed on the ring, it jumps to a figure of 8-which is a simple version of the interwound form. In this case the circular form is stable on account of a local minimum in the total strain energy, even though the interwound form has aglobal minimum at a lower level. When U k is increased further to a critical value of about 3, the circular form ceases to be stable even against small perturbations, and the interwound form is the only configu- ration of stable equilibrium. Le Bret6 has shown that this critical value of a L k is equal to &EIIGJ, which has a value of just over 3 for a square rubber rod. The perturbation mode by which this critical condition may be obtained in the context of the small-strain linear theory of elasticity involves a coupled “ovalization” and “warping” of the ring.

Le Bret6 has estimated that the value of U k a t which a circular hoop of DNA becomes unstable is equal to 6. This follows from his estimate that the value of EIIGJ for DNA is 2 f i (i.e., about double the corresponding value for an ordinary rubber rod) from studies of the Brownian motion of DNA.I4

The mechanics of the interwound form, and its low state of strain energy, can be described most simply as follows. Imagine that the highly twisted circular hoop is grasped by two hands a t opposite ends of a diameter and then pulled out into an elongated oval with two long straight sides. In this configuration the hands must supply a torque to the loop, equal to twice the uniform torque in the circular form (which was entirely self-balancing in that symmetrical configuration); and when one end is released the ab- sence of this torque winds the two straight portions of the oval around each other so that they end up being wrapped together like the two strands of an electric flex, with only small values of curvature and twist. See Cam- erini-Otero and Felsenfeld15 for a discussion of various aspects of the in- terwound state.

Now if the mechanical properties of DNA were those of a uniform lin- ear-elastic rod, we would therefore expect to find in closed loops only the configurations of Fig. l(a) and (c). Benham5 has argued that the toroidal form is a possible equilibrium configuration for a three-dimensional “elastic line”; but Le Bret6 has now shown analytically that the configuration is mechanically unstable, which is indeed in accordance with observations made on physical models.

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The existence of the toroidal form (which I take to be proven) thus poses a paradox. This I now propose to resolve. The key lies in the hypothesis that the mechanical properties of DNA are those of an ordinary linear- elastic rod.

ELASTICITY OF DNA

Levittl6 has done a detailed computer analysis of the elastic response of a length of DNA to an imposed uniform curvature and twist, in connec- tion with the winding of DNA around histone “spools.” Briefly, he models the helical backbones of DNA in terms of atoms with linear-elastic con- straints on bond angles and interatomic distances; and he puts in the bases by means of planar units with, inter alia, highly nonlinear van der Waals interactions with each other. The system is entirely elastic, but since it contains some nonlinear elements, the gross elastic properties of the rod, as deduced from the calculations, must show some nonlinearity also.

In order to understand Levitt’s results in the present context, it is nec- essary to discuss briefly some of the properties of nonlinear elastic systems. Essentially, we are concerned with the nonlinear functional relationship between given bending moment M and twisting moment T applied to a short length of a rod, and the resulting changes of curvature K and twist 7.

The most convenient scheme is to consider the form of the contours of strain energy U in a Cartesian K,T space.

In general, we have, for a quasistatic nonlinear elastic system,

Consequently, the outward-directed normal to the U-contour passing through a given point K,T gives the direction of the corresponding M,T in a Cartesian space whose origin is at the point K,T and whose axes M,T are parallel to K , T , respectively.

It follows immediately that a rod which under the application of pure torque undergoes zero change of curvature has U-contours that intersect the 7-axis orthogonally. Similarly, a rod which under the application of a pure bending moment undergoes zero change of twist has U-contours that intersect the K-axis orthogonally.

Figure 3(a) illustrates both of these features in a rod that happens to be linear-elastic; and indeed for such a rod, each of these features is a direct consequence of the other, since the orthogonality of the U-contours to both axes follows from the absence of a cross-term T K in the general quadratic expression U(.r,~)-see Eq. (3). This is an example of Maxwell’s reciprocal theorem for linear-elastic systems.’l

On the other hand, for a nonlinear elastic rod the orthogonal intersection of the U-contours with one axis does not imply orthogonal intersection with the other axis. Thus, for the kind of system studied by Levitt, the appli- cation of a pure torque does not produce a change of curvature, for reasons

ELASTIC SUPERCOILING OF DNA 1711

of symmetry alone. But on the other hand, Levitt’s results show clearly that the application of a pure bending moment produces not only a change of curvature, but also a change of twist. In other words, the U-contours for DNA must have the general features indicated in Fig. 3(b). In partic- ular, the mechanical symmetry of the system demands only that each complete U-contour be a mirror-image of itself in the 7-axis. Since Levitt16 gives U(K,T) for several values of 7, but only two values of K , it is not possible to plot contours in detail; but it is nevertheless reasonable to be confident about the general features. The cross sections U ( 7 ) at the two values of K have the same parabolic shape; but the functional relationship U ( K * ) , where K* is the maximum dimension of the contour in the K-direction [Fig. 3(b)], is not clear. In general, it must be presumed that the shape of the contours varies somewhat with the value of U.

For present purposes, however, the key feature is simply that the contours intersect the curvature axis nonorthogonally. It follows immediately [Fig. 3(c)] that in the context of the operation depicted in Fig. 2, there is (for a DNA rod, in contrast to a linear-elastic rod) a configuration of minimum strain energy which lies somewhere between the extreme positions of Fig. 2(b) and (c), with the rod adopting the form of a uniform helix. The pitch angle of this helical curve is given by the construction shown in Fig. 3(b). In other words, the “skewness” of the U-contours [in relation to those of the linear-elastic rod, Fig. 3(a)] enables the DNA rod to adopt a helical form in a configuration of stable equilibrium.

The easiest way of visualizing the mechanical properties of a DNA rod is to note that when it is constrained into a tight curve, it occupies a lower energy state if it is also twisted: there is a coupling between curvature and twist whereby a helical form involves less strain energy than a purely cir- cular one. Thus we can see that a closed loop of DNA with, say, A L k = 5 will, under the effect of a sufficiently large perturbation, jump not into an interwound configuration but into a toroidal form as in Fig. l(b).

By an extension of this idea, I suggest that when the amount of dye is increased, the toroidal form will in general be retained, but the number of turns on the toroid will increase. The toroidal form is stable on account of the preference of the DNA to adopt locally a near-helical configuration: each turn of the toroid roughly corresponds to the section PQ of the strip in Fig. 2.

Nevertheless, when the number of turns is sufficiently large, it is possible for a form of instability to occur, in which the plane circular center line of the toroid itself is unstable with respect to small perturbations. This is, conceptually, precisely the same as the situation which we have described above, except that in place of a simple rod, we now have a helical coil whose gross elastic resistance to overall incremental bending and twisting are the relevant parameters. This form of instability must be expected to occur when the number of turns on the toroid exceeds a certain value. When this happens, the toroid begins to change from a plane to a warped configura- tion, and it must therefore end up in something like the interwound form.

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Configuration experiments on rubber rods show that the ordinary in- terwound form is stable against a wide range of perturbations, and that so also is the Y-shaped interwound form of Fig. l(c), into which it may be maneuvered. It therefore seems improbable that transformations will occur easily between them unless, possibly, there is an extremely low ef- fective coefficient of friction. It seems likely, therefore, that the final form reflects events which occur immediately after the toroidal form has become unstable. It is conceivable that in the process of jumping to the final configuration, the tightly curved end loops may be predisposed to occur a t stretches in the DNA, which, on account of particular sequences of base pairs, are locally less stiff in flexure than other parts of the rod. “Weak zones” of this sort could conceivably “steer” the DNA in the dynamic phase into a Y-shaped form shown in Fig. l(c). Any further increase in the amount of dye absorbed would increase the number of turns of the inter- wound stretches and tighten the end loops.

DISCUSSION

The picture which emerges from the analysis may be summarized as follows. When an increasing amount of dye is introduced to a relaxed loop of circular DNA, the loop at first stays circular, but then it reaches a point a t which the circular form is no longer stable. The loop changes its con- figuration to a toroidal form which is stabilized by the peculiar nonlinear elasticity of a DNA rod, in which there is a coupling between bending and twisting. As the amount of dye is increased further, the number of coils on the toroid steadily increases, until a point is reached at which the toroidal form itself becomes unstable, and the loop of DNA jumps into an inter- wound configuration.

Le Bret’s6 estimate of the critical value of ALk at which the circular form becomes unstable agrees well with studies of sedimentation analyses, which associate the toroidal form with A L k = 5. The value of A L k at which the toroidal form becomes unstable in favor of the interwound form would be much more difficult to estimate, even if the nonlinear elastic properties of DNA were known precisely.

I thank Drs. A. Klug, M. Levitt, and W. R. Bauer for advice and comments.

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Received August 13,1979 Accepted March 19,1980