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Circular and helical equilibrium solutions of inhomogeneous rods

Real filaments are not perfectly homogeneous. Most of them have various materials composition and shapes making their stiffnesses not constant along the arclength. We investigate the existence of circular and helical equilibrium solutions of an intrinsically straight rod with varying bending and twisting stiffnesses, within the framework of the Kirchhoff model. The planar ring equilibrium solution only exists for a rod with a given form of variation of the bending stiffness. We show that the well known circular helix is not an equilibrium solution of the static Kirchhoff equations for a rod with non constant bending stiffness. Our results may provide an explanation for the variation of the curvature seen in small closed DNAs immersed in a solution containing Zn^{2+}, and in the DNA wrapped around a nucleosome.


Introduction
Static and dynamics of one-dimensional structures have been the subject of intense research in Engineering, Physics and Biology. Such filamentary objects are present in both nature and man-made devices. The stability of sub-oceanic cables [1,2], the tridimensional structure of biomolecules [3] and bacterial fibers [4], and the mechanical properties of nanosprings [5,6,7,8] are few of many interesting problems investigated within elastic rod models.
In most cases, the rod is considered as having constant or uniform stiffness, but non-uniform rods have been considered in the literature. It has been shown that nonhomogeneous Kirchhoff rods may present spatial chaos [9,10]. Domokos and Holmes studied buckled states of non-uniform elasticae [11].
Deviations from the helical structure of rods due to periodic variation of the Young's modulus were verified numerically by da Fonseca, Malta and de Aguiar [12]. Homogeneous and nonhomogeneous rods subject to given boundary conditions were studied by da Fonseca and de Aguiar in [13]. The effects of a nonhomogeneous mass distribution in the dynamics of unstable closed rods have been analyzed by Fonseca and de Aguiar [14]. Goriely and McMillen [15] studied the dynamics of cracking whips [16], Kashimoto and Shiraishi [17] studied twisting waves in inhomogeneous rods, and Neuringer and Elishakoff [18] studied natural frequencies of inhomogeneous rods.
It is well known that straight, circular and helical rods are equilibrium solutions of the Kirchhoff model [19,20,21]. These solutions have been used to study the stability and buckling phenomena of loop elasticae [1,22], the twist to writhe conversion phenomenon in twisted straight rods [23], and the stability of DNA configurations with or without self-contact [24], to name some examples.
Our aim here is to investigate the existence of circular and helical equilibrium solutions for an intrinsically straight, inextensible, unshearable, and inhomogeneous rod with circular cross-section, within the Kirchhoff rod model.
Establishing the stability (even in the case of simple equilibrium configurations) of a rod is a non-trivial task (see, for example, Refs. [22,23,24,25,26,27,28]). Since the study of the stability analysis departs from the knowledge of equilibrium solutions, our work is the starting point for the study of the stability of equilibrium solutions of inhomogeneous rods.
The inhomogeneity of the rod is considered through the bending and twisting coefficients varying along its arclength s, B(s) and C(s), respectively.
We shall derive a set of non-linear differential equations for the curvature, k F (s), and the torsion, τ F (s), of the centerline of an inhomogeneous rod, and then impose the necessary conditions to finding circular/helical solutions. Circular (ring) solutions have k F = Constant and τ F = 0, while helical solutions have the curvature and torsion of the rod centerline satisfying the Lancret's theorem: k F (s)/τ F (s) is constant [29,30]. The well known circular helix is the particular case where both k F and τ F are constant.
The fundamental theorem for space curves [30] states that the curvature and the torsion completely determine a space curve, but for its position in space. This is the reason for our choice of working with the curvature, k F (s), and the torsion, τ F (s), of the rod centerline. We shall show that the solutions for k F (s) and τ F (s) depend only on the bending coefficient, B(s), an expected result since the centerline of the rod does not depend on the twisting coefficient (see for instance, Neukirch and Henderson [31]).
In Sec. II we present the static Kirchhoff equations for an intrinsically straight rod with circular cross section and varying stiffness, and derive the non-linear differential equations for the curvature and torsion of the rod. In Sec. III we analyse the cases of null torsion (straight and planar rods). We where the angle ξ is the amount of twisting of the local basis with respect to t.
We are interested in the equilibrium solutions of the Kirchhoff model, so our study departs from the static Kirchhoff equations [32]. For intrinsically straight isotropic rods, these equations are: where the following scaled variables were introduced: the vectors F and M are the resultant force and the corresponding moment with respect to the centerline of the rod, respectively, at a given cross section.
The prime ′ denotes differentiation with respect to s. k i are the components of the twist vector, k, that controls the variations of the director basis along the rod through the relation k 1 and k 2 are related to the curvature of the centerline of the rod (k F = k 2 1 + k 2 2 ) and k 3 is the twist density. B(s) and C(s) are the bending and twisting coefficients of the rod, respectively. Writing the force F in the director basis, the equations (2-4) give the following differential equations for the components of the force and twist vector: The equation (13) shows that the component M 3 = C(s) k 3 of the moment in the director basis (also called torsional moment), is constant along the rod, consequently the twist density k 3 is inversely proportional to the twisting In order to look for circular (ring) and helical solutions of the Eqs. (8)(9)(10)(11)(12)(13) the components of the twist vector k are expressed as follows: where k F (s) and τ F (s) are the curvature and torsion, respectively, of the rod centerline, and ξ is given by Eq. (1).
The Eqs. (18)(19)(20) for the curvature, k F , and torsion, τ F , do not depend on the twisting coefficient, C(s). Therefore, the centerline of an inhomogeneous rod does not depend on the twisting coefficient unlike the case of homogeneous rods (see, for example, Eqs. (13) and (14) of Ref. [31]).
Langer and Singer [33] have obtained a set of first-order ordinary differential equations for the curvature and torsion of the centerline of a homogeneous rod that contains terms proportional to k 2 F and τ 2 F . The Eqs. (18)(19)(20) have the advantage of involving only terms linear in k F and τ F .
where we have set k F = K, and C 0 is a constant of integration. Therefore, the twisted (M 3 = 0) ring solution of an inhomogeneous rod only exists if its bending stiffness satisfies the differential equation (25) (forced harmonic oscillator). The bending stiffness (solution of (25)) can be written in the form: with A 0 and B 0 arbitrary constants.
Differentiating f 3 with respect to s and substituting in Eq. (23), we obtain the following non-linear differential equation for the curvature k F (s): If k F = Constant ≡ K (the planar ring solution) the above equation gives the following differential equation for B: which is the same Eq. (24) and, therefore, gives the same condition for B(s) shown in Eq. (26). In this case, f 3 (s) is given by: So, the non-twisted planar ring (k F = Constant) is also an equilibrium solution of the static Kirchhoff equations if the bending stiffness of the inhomogeneous rod is given by Eq. (26).
Another solution of Eq. (28) for the curvature k F (s) comes from making (Bk F ) ′ = 0, which gives and, from Eq. (27)

Helical solutions of inhomogeneous rods
In this section, we are considering the equilibrium solutions for the rod centerline in which k F = 0 and τ F = 0. In order to find helical solutions for the static Kirchhoff equations, we apply the Lancret's theorem to the equations (18)(19)(20). We first write the Lancret's theorem in the form: where β = 0 is a constant. Substituting Eq. (32) in Eq. (18) we obtain Differentiating f 3 with respect to s and substituting in Eq. (20) we obtain the following differential equation for τ F (s): where Eq. (33) was used to simplify the above equation. One immediate solution for this differential equation is that substituted in Eq. (33) gives For non-constant τ F , the Eq. (37) gives the following solution for τ F : If M 3 = 0, the only solution satisfying the Lancret theorem is τ F (s) = 0 which is the planar solution given in the previous section. Substituting Eqs.
By choosing the z-direction of the fixed cartesian basis as the direction of the helical axis, we can integrate t in order to obtain the three-dimensional configuration of the centerline of the rod.  that k F (0) and M 3 are free parameters that have been chosen so that the helical solutions displayed in Fig. 2

Conclusions
The existence of circular (planar ring) and helical configurations for a rod with non-constant bending and twisting stiffnesses has been investigated within the framework of the Kirchhoff rod model. We are interested in finding analytical solutions, and since the static Kirchhoff equations consist of nine non-linear PDEs, we derived from them a set of differential equations (Eqs. (18)(19)(20)) for the curvature, k F , and torsion, τ F , of the rod centerline, and then applied the geometric conditions to obtain circular and helical so-lutions.
We have shown that the planar ring (k May an intrinsically straight homogeneous elastic rod present other types of generalized helical equilibrium solutions ? A homogeneous rod has C constant, implying k 3 = Constant (from Eq. (13)). It has been proved that ξ ′ = 0 for a helical solution of a homogeneous rod (see reference [35]), and Eq. (17) shows that the torsion τ F = k 3 = Constant. In order to satisfy the Lancret's theorem (Eq. (32)), the curvature k F of this helical solution must also be constant. Since the helical structure whose curvature, k F , and torsion, τ F , are constant, is a circular helix, the only type of helical solution for an intrinsically straight homogeneous rod is the circular helix.
Some motivations for this work are related to defects [36] and distortions [37] in biological molecules. Also, the stiffness of the DNA molecule has been proved to be sequence-dependent [3]. These defects, distortions and sequence-dependent elastic properties could be modeled as inhomogeneities along a continuous elastic rod.
Haijun and Zhong-can [34] have used the Goriely and Tabor stability analysis method [22,23] [39]. Our results suggest that the non-circular shape of the 126-bps closed DNAs is a consequence of the binding of Zn 2+ ions to the basepairs that decreases locally the stiffness of the DNA, thus increasing the curvature.
It has been proposed [40,41,42] that the kinking phenomenon is an important mechanism for wrapping DNA around nucleosomal proteins. In 1997, Luger et al [43] reported the X-ray crystalline structure of the nucleosomal protein, demonstrating that the DNA is not uniformly bent around nucleosome, but presents minimum and maximum curvatures at different positions.
It is well known that the stiffness of DNA is sequence-dependent. Our results show that the variation of the bending stiffness of the DNA can give rise to these minimum and maximum curvatures.
First, we extract f 1 and f 2 from Eqs. (55) and (54), respectively: where M 3 = C(s) (ξ ′ +τ F ) is the torsional moment of the rod that is constant by Eq. (56). Differentiating f 1 and f 2 with respect to s, substituting in Eqs.