Geometry of the toroidal N-helix: optimal-packing and zero-twist

Two important geometrical properties of N-helix structures are influenced by bending. One is maximizing the volume fraction, which is called optimal-packing, and the other is having a vanishing strain-twist coupling, which is called zero-twist. Zero-twist helices rotate neither in one nor in the other direction under pull. The packing problem for tubular N-helices is extended to bent helices where the strands are coiled on toruses. We analyze the geometry of open circular helices and develop criteria for the strands to be in contact. The analysis is applied to a single, a double and a triple helix. General N-helices are discussed, as well as zero-twist helices for N > 1. The derived geometrical restrictions are gradually modified by changing the aspect ratio of the torus.


Introduction
The effect of bending on the geometry and characteristics of helices of tubes with nonzero thickness is described. In particular, we explore optimization of the volume fraction, called optimal-packing in analogy to the terminology used for sphere packings, and we explore structures with zero strain-twist coupling as for ideal ropes. Our aim is to understand better the geometrical constraints on, and the behavior of, helical molecules such as biomolecules, carbon nanotubes and polymer nanofibers.
Helical biomolecules can sometimes, but not always, be described as a straight helix. For example, DNA is often bent when present in a molecular complex, e.g. with proteins and in chromatin. Therefore, it is important to understand the bending of an N -stranded helix from a geometrical point of view. The amount of bending in an N -helix is described mathematically by coiling the helix on a torus with aspect ratio a/R 1, where the helix radius is a and the torus radius is R. For small aspect ratio the N -helix is only slightly bent; for larger ratios, significant bending is present. Certain generic properties depend on the amount of bending: optimal-packing (or best packing, defined as a structure that maximizes a volume fraction), zero-twist (a structure that behaves under pull as a non-chiral structure; it rotates neither in one nor in the other way) and winding (a structure that rotates counter to unwinding under a pull).
Theoretical work on bent biomolecules, like the bending of the double helix of DNA, has largely gone in one direction. It has been assumed that deformation phenomena regarding DNA operate on a scale at which the internal double-helix structure is mostly irrelevant, and a large body of literature deals with mathematical and mechanical aspects of bent, coiled and supercoiled DNA, where the double helix of DNA is represented as a single flexible tube. An early elastic model of supercoiling in DNA was suggested by Benham [1]. Jülicher [2] has investigated configurations of closed infinitely thin rods with self-contact to describe a phase 3 diagram of supercoiling. For rods of nonzero diameter, Stump et al [3] used an approximate method for calculating the configurations where lines of contact are present. The influence of end-conditions on self-contact in DNA loops has been studied by Tobias et al [4] and Coleman et al [5]. The statistical mechanics of supercoils has been studied by Marko and Siggia [6]. The stability of plasmids has been studied by Tobias et al [7] and Coleman et al [8]. For a detailed description of such single-tube models, see also Coleman and Swigon [9] and Travers and Thompson [10]. Further, in Thompson et al [11], the mechanics of ply formation in DNA supercoils has been studied. For a review of DNA mechanics, see Benham and Mielke [12]. Recently, Zheng et al [13] have argued that the elastic rod model of DNA is insufficient for longer segments of DNA. A double-strand elastic theory of DNA has been given by Moakher and Maddocks [14]. Equilibrium shapes for flat knots of simple polymers have been investigated from the viewpoint of scaling by Metzler et al [15].
In the literature, elastic rod models have been used to calculate the dependence of DNA configurations on the linking number, Lk. This is not a trivial task, mainly because the rods have nonzero diameter. It follows from the theorem related to the linking, see Cǎlugǎreanu [16], Pohl [17], White [18] and Fuller [19], that the topological constant Lk obeys the relation in which T w and W r are the twist and writhe of the (closed) curve. Although Lk is a topological invariant integer, W r and T w are not and depend on geometry. The configurations of interest for computing Lk are supercoiled configurations where the DNA makes contact with itself. Tubular models with one or more strands have been used, for example, to describe motifs of biological chain molecules, and thick knots that are of mathematical interest. Aspects of the geometry of straight tubular helices have been studied by Przybył and Pierański [20], Neukirch and van der Heijden [21], Gonzalez and Maddocks [22], Maritan et al [23] and Stasiak and Maddocks [24]. The importance of entropy for helix formation has been examined by Snir and Kamien [25]. For various applications of tube models to biomolecules, see Banavar and Maritan [26] and Banavar et al [27], and for applications to knot theory, see Przybył and Pierański [28]. According to Gonzales and Maddocks, the thickness of a knot can also be defined in terms of a global radius of curvature [29]. Knots made with as little 'rope' as possible have been designated ideal knots [38]. In an earlier work, we looked at the tubular single and double helices [30]. Optimization of the volume fraction was shown to be consistent with many of the common motifs of the molecular structures, e.g. in α-helices and DNA. In addition, a new possible motif for collagen was suggested on the basis of packing and twisting properties of a unique triple helix [31]. The study of optimal volume packing for a double helix was recently revisited by O'Hara [32] and is in agreement with earlier results for single and double helices [30].

The circular helix
A helical helix is described by x(t) = (R + a cos(ωt)) cos t, y(t) = (R + a cos(ωt)) sin t, z(t) = bt + a sin(ωt) (2) for t ∈ R and a, b, ω, R are positive constants. If a = 0 and b = 0, as we assume in the following, then equations (2) describe a helix wrapped around a torus. We call this wrapping a in the x-z plane. In this plot, the circular helix has the aspect ratio a/R = 0.1. circular helix; later we will discuss the case of circular N -helices, where N is the number of strands. The torus has inner radius R − a and outer radius R + a; its aspect ratio is a/R. For the center line, the radius of curvature is R, see figure 1. For packed tubular helices, the straight helix (a helix coiling on a straight cylinder) should intuitively be obtained in the limit a/R = 0. One can define a torus pitch, H , as the length of repetition along the center line; for an arc of radius R, it must be H = 2π R/ω. The corresponding reduced pitch is h = H/2π = R/ω. Note that when ω is an integer the helix makes a closed loop, and ω becomes the number of times the helix coils around the torus.
To make the comparison with the straight helix (a/R = 0) transparent, we use an alternative parameterization of the circular helix. We redefine t → t * = ωt, and set ω = R/ h. The following parameter equation gives a straight helix of radius a and reduced pitch h in the z-direction in the limit h/R → 0: For the straight helix, the helix line makes a constant angle, called the pitch angle v ⊥ , with the horizontal plane. This is not the case with a circular helix, where and the derivative squared is The tangent of the circular helix makes an angle, v, with the center line of the torus. It follows from (5) that this angle is determined by The angle v is therefore not constant in t; it has a single minimum in the interval [0, 2π] and becomes approximately constant in the limit of small bending. It approaches the constant value 90 where v ⊥ is the pitch angle for a straight helix. Two important parameters for the following discussion are the relative pitch, h/a, measuring the repetition length of the circular helix and the aspect ratio, a/R, which measures the amount of bending of the helix.
We now investigate self-contacts in a tubular circular helix. For self-contacts in the straight helix, this packing problem is solved by a continuous trace of points. For the circular helix given by equations (2) there is only a discrete set of points with self-contact, as can be seen in figure 2. We assume that the tube helix has hard walls around the helix line. The cross-sections of the tubes are circular and the diameter is D. For packing to make sense, there are two conditions, one local and the other global in nature, that must be satisfied by the tube structure. The global condition is that the tubes are in contact at the most restrictive geometry, i.e. on the plane of the torus pointing inwards. The local condition is that the backbone tube volume should be preserved along the center line of the tubes. The backbone tube volume is preserved if D 2/κ, where κ is the local curvature of the helix line. The local condition is that the radius of the tube is smaller than or equal to the radius of curvature. When this Poisson criterion is obeyed, the volume of a tubular helix is π D 2 L/4, where L is the curve length of the helix.
The global condition of contact between tubes is found out in two steps: firstly, by considering the set of connecting points on the helical line for which the line through the two points is perpendicular to the helical line; secondly, the distance between these two points should be equal to the tube diameter. Now, take two arbitrary points on the circular helix: and 6 The square of the distance between the two points is With the chosen parameterization, the contact point for the most restrictive solution is (x, 0, 0), where x is negative. The two points in contact with each other of the circular helix are in a symmetric relation to each other around π , i.e. as the parameter along the helix we use . Then equation (9) becomes The derivative of the distance squared reads

Optimal-packing in a circular helix
The optimally packed circular helix is defined as the one with the highest volume fraction for the space occupied by the tubular structure, under the condition that the bending is constant, i.e. that the torus aspect ratio is constant. For the straight helix (a/R = 0) the optimally packed helix is also called the close-packed helix [30]. For a configuration of a tube with no self-contact, there will always be a more densely packed structure with self-contacts. We therefore consider circular helix confirmations with self-contact in the following. What is the condition for self-contact at the most restrictive configuration? Given an aspect ratio a/R, we find the extremums for D 2 1 , the interpoint distance squared along the helical line, where D 1 = D, i.e. for D 2 1 = 4a 2 sin 2 t + 4(R + a cos t) 2 sin 2 (ht/R).
The condition d dt D 2 1 = 0 for the extremum at t can be written as The solutions, i.e. the map of perpendicular points, of this transcendental equation are found numerically for various values of the aspect ratio, a/R, and are found to be well behaved in the limit of small bending. The map of perpendicular points, equation (13), are shown in figure 3; obviously, solutions are symmetric under reflection symmetry t → −t. For zero bending, a/R = 0, the map of perpendicular points is shown in figure 3(a). On the right-hand side of the first hairpin (t > 0) are the solutions with self-contact. The maximal value of the relative pitch for self-contact is h/a = 0.466. For the second hairpin, the two perpendicular points are further separated from each other by an additional length of the helix and so on for consecutive hairpins. When bending is introduced, new solutions extending to infinity appear; see left-and right-hand sides of figure 3(b). These solutions result from the merging of an upwards-and downwards-pointing hairpin. The branch at t = 0 is the trivial solution where the two connecting points coincide. The volume fraction for the tubular helix is defined as the ratio of the tubular volume to a reference volume, We take the reference volume V E to be the volume of a piece of a smallest enclosing torus. We have V E = π(a + D/2) 2 H . The corresponding volume of the tubular helix is Note that with increased bending, the function f V becomes nearly flat around its maximum.
The helical volume fraction can be written as a function of h/a and a/R: The volume fraction, f V , is plotted as a function of h/a for three different aspect ratios in figure 4. For zero bending, i.e. a/R = 0, the maximum of f V is at h/a = 0.328, where the volume fraction is f * V = 0.784. It is the close-packed single helix in [30]. The maximal volume fraction decreases on further bending. For example, when a/R = 0.1 we find that f * V = 0.717. Furthermore, the optimal relative pitch, h/a, where f V is maximal (CP structure) and the single helix is optimally packed, is an increasing function of the aspect ratio; figure 5 shows how the optimal relative pitch, h/a, depends on the aspect ratio a/R.

Twisting-behavior of the circular helix
In this section, we discuss how a circular helix with inter-tubular contacts behaves under strain. Let us look at a strand of the circular helix of length 2π in the parameter t; the bending of the helix implies that this strand wraps a circle of radius R. Then the total twist, , is observed The optimal relative pitch h/a as a function of the aspect ratio a/R for a circular helix. The red solid curve is for the optimally packed circular helix. At zero bending, a/R = 0, the optimal relative pitch is 0.328 (CP). The curve is plotted in the interval a/R 0.235. For higher aspect ratio (at approximately a/R > 0.24), there is no optimal relative pitch with the usual restrictive contact, the h/a value corresponds to a point not located on the first hairpin in figure 3.
The twist angle is measured between a vector pointing from the circular center line of the torus to the helix line, and the plane of the torus. The angle v ⊥ = 90 • − v is determined by equation (6). The curve length of this strand is To measure how the helix behaves under strain, the following function is introduced. The incremental twist, f , is the dimensionless ratio of twist to length multiplied by the diameter of the tube, D, i.e. of the individual strands and are as such unphysical [30]. In the next section, dealing with the circular double helix, we will encounter the zero-twist structures at the termination of winding.

Optimal-packing in a circular double helix
What is the condition for the two tubes touching at the most restrictive configuration? The condition for the extremum of D 2 2 is and the solutions, i.e. perpendicular points on different strands, are plotted in figure 8.
For the branch at t = −π/2, the phase difference is π and the two points opposite to each other are in the same equatorial plane. For high pitch this becomes the minimal distance. For lower pitch the distance along this branch is not minimal. At zero bending ( figure 8(a)), there is a straight line of contact when t = −π/2. On the right-hand side of the central hairpin, there is a helical line of contact. The maximal value with a helical line of contact is h/a = 1. In this case, the most restrictive packing is achieved by following the central hairpin from t = 0, moving left to the maximum at t = −π/2 and then following the straight line to infinity. In the second hairpin the two perpendicular points are further separated from each other by an additional length of the helix, and so on for consecutive hairpins. When bending is introduced, new sets of perpendicular points appear; see the downwards-pointing hairpin in figure 8(b). Upon further bending, upwards-and downwards-pointing hairpins can nest and extended solutions can cross over, as has happened in figure 8(c). This phenomenon is frequently observed in the evolution of these maps.
The volume of the two tubes becomes and the volume fraction as a function of a/ h and a/R is In figure 9, the volume fraction, f V , is plotted as a function of h/a. For zero bending (solid line) the maximum of f V is at h/a = 0.636, where the volume fraction is f * V = 0.769. It is the close-packed double helix of [30]. The optimal packing depends on the amount of bending, i.e. on the aspect ratio a/R: the maximum of f V is a decreasing function of a/R; see the discussion below figure 9.

Zero-twist of the circular double helix
We now discuss how a circular double helix with inter-tubular contacts behaves under strain. In figure 10, the incremental twist f is plotted as a function of h/a for four values of the aspect ratio a/R. For zero bending (solid line) the maximum of f is at h/a = 0.821, where f * = 1.478. This is the zero-twist structure of a straight double helix with d f = 0. One sees that for small values of h/a the incremental twist f is an increasing function and therefore the circular double helix will wind-up under strain (rotate counter to unwinding); at larger values of h/a it is a decreasing function and unwinds as expected. With bending, there is generally a nontrivial maximum of f and therefore a zero-twist structure. The zero-twist structures for straight helices are described in [37,39].

The optimally packed double helix becomes a zero-twist structure under bending
Let us systematically investigate how the optimally packed (CP) and zero-twist (ZT) structures depend on the aspect ratio a/R in a circular double helix. In a straight helix, i.e. for a/R = 0, the close-packed structure and zero-twist structures have h/a values of 0.636 and 0.821, respectively. The two properties, being close-packed or being zero-twist, cannot be satisfied at the same time. This is not true when bending is allowed. Figure 11 depicts the values of h/a for the optimally packed double helices (red dotted line) and for the zero-twisted double helices (blue solid line) as a function of bending. The pitch of the optimally packed helix increases more rapidly than that of the zero-twist helix upon increasing bending. At an aspect ratio of a/R = 0.201, the circular double helix is both zero-twisted and optimally packed. The closed circular double helix in figure 7(b) has a/R = 0.20, so it is approximately such a structure.

The circular triple helix
The general symmetric circular triple helix has a parametric equation for t 1 , t 2 , t 3 ∈ R. We use a parameterization where t ≡ t 1 = −(t 2 + 2π/3). Now, the square of the distance between a point starting on one helical line and ending on the next becomes D 2 3 = 4a 2 sin 2 t + 4(R − a cos t) 2 sin 2 (ht/R + hπ/3R) and the derivative of D 2 3 reads d dt The corresponding maps of perpendicular points are shown in figure 12.

Optimal-packing and zero-twist in a circular triple helix
What is the condition for tubes touching at the most restrictive condition, that is, D 3 = D? The condition for the extremum at t can be written as Then, we find for the volume fraction, f V , as a function of a/ h and a/R: This volume fraction has been plotted in figure 13 for four values of the aspect ratio a/R. For zero bending (solid line) the maximum of f V is at h/a = 0.943, where f * V = 0.744. In figure 14, the accompanying rotation per unit length, f , is plotted for the triple helix as a function of h/a for four values of the aspect ratio. Now, the maximum of f is at h/a = 0.927, where f * = 1.022. Figure 15 is a plot of the optimal h/a for optimally packed and zero-twist circular helices as a function of the aspect ratio a/R. The optimal relative pitch h/a is approximately equal for a/R = 0, that is, a straight helix: h/a = 0.943 and 0.927 for optimally packed and zerotwist, respectively. Further bending makes the two lines separate; the optimally packed structure always has a larger relative pitch than the zero-twist structure. Therefore, the triple helix cannot become both optimally packed and zero-twist by additional bending, and is only approximately so at zero bending. We have suggested this to be relevant for understanding the mechanics of the triple helix of collagen [31]. Here the relative pitch, h/a, is plotted as a function of t. The branch starting at t = 0, and extending to infinity, represents the most restrictive solution for contact.

The general N-helix
What happens at larger N ? In figure 16, we have plotted the optimal-packing and zero-twist curves for the case of N = 4. It is observed that at a/R = 0 the relative pitch for optimal-packing is h/a = 1.362. At N 5, the location of the optimally packed structure moves to infinity. This corresponds to the pitch angle being v ⊥ = 90 • , i.e. that the tubes are straight and parallel (as tanv ⊥ = h/a). For the straight N -helix, the zero-twist angle moves towards v ⊥ = 45 • (h/a = 1) for large N and its relative pitch, h/a, is always less than one. Likewise, for the bent N -helix, the zero-twist line changes rapidly with N for N = 2, 3 and 4. For N > 4, the zero-twist lines display only a little change with N . At large N , the solutions for N or N + 1 strands become nearly identical. For the overall trend with increasing N of optimal-packing and zero-twist, compare figures 11, 15 and 16.

Discussion and conclusion
The tubular geometry of open and closed toroidal helices is investigated using differential geometry and numerical calculations. A particular requirement is that the tubes do not intercept The circular quadruple helix. The relative pitch h/a as a function of the aspect ratio a/R. The red dotted curve is for the optimally packed quadruple helix and the blue solid curve for the zero-twist quadruple helix; at zero bending, a/R = 0, the relative pitch is 1.362 (CP) and 0.960 (ZT), respectively. With more than N = 4 strands, the optimal-packing line will move to infinity; the zero-twist line only moves slightly and approaches a curve starting at h/a = 1.
The effectiveness of the local packing is described by a volume fraction and when maximized we define these structures as being the optimally packed toroidal helices. The twisting properties of the toroidal helices are also derived and the zero-twist structures found in the case of multiple strands. For double-stranded circular helices, there exists one that is Table 1. Optimal-packing of circular single, double and triple helices: a/R is the aspect ratio of the torus that the helix coils on and measures the amount of bending, h/a is the relative pitch, i.e. the ratio of reduced helix pitch to helix radius, 2a/D is the ratio of the helix diameter and the diameter of the helical tubes, f * V is the volume fraction of the straight helix and f V / f * V is the relative volume fraction compared to that of the straight helix. The straight double helix of DNA is not a zero-twist structure, but is close-packed [37]. However, it follows from the analysis of the twisting behavior that molecular bent double helices, such as DNA, might behave as zero-twist structures under certain conditions. This can be important for coiling of DNA (or RNA) when it is subject to strain, as otherwise it would twist under strain.
As the twisting and packing properties change fairly rapidly with a/R, such properties can become determining for the specific geometry of molecular complexes. Perhaps it will be advantageous for molecular structures, such as the chromatin fiber, to be both optimally packed and zero-twist. The usual B-form of DNA is not a symmetric double helix; therefore, for the geometry of the chromatin fiber one needs to carefully model the asymmetric double helix as was done for the straight helix in [30].
For triple helices the straight helix is a structure that is approximately optimally packed and zero-twist at the same time; with bending, the solutions that are optimally packed and zerotwisted begin to separate further. These results are likely to be relevant for understanding the coiling of chiral structures in nature, specifically the difference between the coiling of double helices (as DNA) and triple helices (such as collagen and some polysaccharides).
The effect of bending on the volume fraction on single, double and triple circular helices is summarized in table 1. It is observed that there is a significant difference in the local packing behavior of the structures, depending on the number of strands. For example, a bending of 10% (a/R = 0.1) of the single helix will reduce the local volume fraction by 17%, and the same amount of bending on the triple helix will only reduce the volume fraction by 7% compared to the straight helix. For molecules where molecular forces are similar, this can be expected to be revealed through differences in persistence lengths.
One of the conclusions of this paper is that the internal double-helical structure of DNA cannot be ignored when modeling complex DNA structures. The analysis given is relevant for closed circular DNA molecules, which occur in some simple biological systems, e.g. plasmids, viruses, see Vinograd and Lebowitz [33] and Clewell and Helinski [34], and is relevant for the coiling of DNA inside the chromosomes. Other examples are packing of RNA in viruses, as in the coronavirus torovirus, which has a toroidal geometry [35], and the horseshoe-shaped TLR3 molecule (toll-like receptor 3, PDB entry 2A0Z [36]) with its near-perfect toroidal helix packing. The analysis might be relevant for the coiling and twisting of carbon nanohelices, as reported in [40][41][42].