Introduction

A biopolymer is often modeled as an elastic filament owing to its chain structure. Conformational and mechanical properties of a filament have attracted a lot of attention for a long time owing to its wide range of applications in either macroscopic objects such as pillars or microscopic objects such as semiflexible biopolymers (Benham 1977, 1989; Tanaka and Takahashi 1985; Fain et al. 1997; Fain and Rudnick 1999; Panyukov and Rabin 2001; Zhou et al. 2005, 2007; Zhou 2018; Love 1944; Marko and Siggia 1994; Bustamante et al. 1994; Marko and Siggia 1998; Kratky and Porod 1949; Panyukov and Rabin 2000; Goriely and Shipman 2000; Kessler and Rabin 2003; Erickson et al. 1996; Srinivasan et al. 2008; Smith et al. 2001; Jones et al. 2001; Iwai et al. 2002; Daniel and Errington 2003; Gitai et al. 2004, 2005; Kruse et al. 2005; Carballido-López 2006; Vats and Rothfield 2007; van der Heijden 2001; Chouaieb et al. 2006; Allard and Rutenberg 2009; Zhou et al. 2017; Jung and Ha 2019; Panyukov and Rabin 2002; Shih et al. 2003; Taghbalout and Rothfield 2007; Thanedar and Margolin 2004; Vaillant et al. 2005; Esue et al. 2006; Andrews and Arkin 2007; Russell and Keiler 2007; Srinivasan et al. 2007; Zhou 2007; Moukhtar et al. 2007; Starostin and van der Heijden 2010; Zhou et al. 2014).

The configuration of a filament can be described by the shape of its centerline and the twist of its cross section around the centerline. Denoting the arc length of the centerline as s and the locus of centerline as r(s), the configuration of a filament can be described by a triad of unit vectors \(\{\mathbf{t}_i\}_{i=1,2,3}\), in which \(\mathbf{t}_3 \equiv \dot{\mathbf{r}}\) is the tangent to the center line, t\(_1\) and t\(_2\) are oriented along the principal axes of the cross section (Benham 1977, 1989; Tanaka and Takahashi 1985; Fain et al. 1997; Fain and Rudnick 1999; Panyukov and Rabin 2001; Zhou et al. 2005, 2007; Zhou 2018) and the symbol “.” represents the derivative with respect to s. \(\{\mathbf{t}_i\}_{i=1,2,3}\) satisfy the generalized Frenet equations \(\dot{\mathbf{t}}_i={\varvec{\omega }}\times \mathbf{t}_i\) (Benham 1977, 1989; Tanaka and Takahashi 1985; Fain et al. 1997; Fain and Rudnick 1999; Panyukov and Rabin 2001; Zhou et al. 2005, 2007; Zhou 2018), and the vector \({\varvec{\omega }}=(\omega _1, \omega _2, \omega _3)\) represents curvature and torsion parameters (Benham 1977, 1989 Tanaka and Takahashi 1985; Fain et al. 1997; Fain and Rudnick 1999; Panyukov and Rabin 2001; Zhou et al. 2005, 2007; Zhou 2018).

Moreover, intrinsic properties and external conditions determine the conformal and mechanical properties of a filament. The intrinsic properties are usually referred to as bending rigidities, twisting rigidity and inertia tensor (Benham 1977; Tanaka and Takahashi 1985; Fain et al. 1997; Fain and Rudnick 1999; Panyukov and Rabin 2001; Zhou et al. 2005, 2007; Zhou 2018; Love 1944). On the other hand, external physical conditions include applied forces, torques and constraints such as boundary conditions (BCs) or confinements. To explore different physical properties, it usually requires different models. For instance, regarding a filament as an inextensible chain with a finite bending rigidity but a zero cross-section area leads to the wormlike chain (WLC) model, and it has been applied to describe the entropic elasticity of some semiflexible biopolymers (Marko and Siggia 1994; Bustamante et al. 1994; Marko and Siggia 1998; Kratky and Porod 1949). Another simple model is the wormlike rod chain (WLRC) model which views a filament as a chain of a finite intrinsic twist and an isotropic cross section (Fain et al. 1997; Marko and Siggia 1994; Bustamante et al. 1994; Marko and Siggia 1998). Both WLC and WLRC models are intrinsically straight, i.e., free of external force and torque, and their ground-state configurations (GSCs, or the configuration with the lowest energy) are either a straight line or a straight cylinder, respectively. In contrast, the unique GSC of an intrinsically curved filament gives a curved centerline with a given curvature when it is free of external force and torque.

A filament can form various structures and the simplest but very useful one is a helix. A helix may result from a finite intrinsic curvature and torsion (Zhou et al. 2005, 2007; Zhou 2018; Panyukov and Rabin 2000; Goriely and Shipman 2000; Kessler and Rabin 2003; Erickson et al. 1996; Srinivasan et al. 2008; Smith et al. 2001). Free of external force or torque, such a filament has naturally a helical shape and can maintain the helix under a uniaxial force (Zhou et al. 2005, 2007; Panyukov and Rabin 2000; Goriely and Shipman 2000; Kessler and Rabin 2003). In contrast, under a uniaxial force to have a helical shape, an intrinsically straight filament requires some external torques or constraints, such as a MreB molecule inside a cylindrical bacteria (Jones et al. 2001; Iwai et al. 2002; Daniel and Errington 2003; Gitai et al. 2004, 2005; Kruse et al. 2005; Carballido-López 2006; Vats and Rothfield 2007). The elasticity of a helix, either in three-dimensional (3D) space or on the surface of a cylinder, has been studied extensively (Zhou et al. 2005, 2007; Zhou 2018; Love 1944; Panyukov and Rabin 2000; Goriely and Shipman 2000; Kessler and Rabin 2003; van der Heijden 2001; Chouaieb et al. 2006; Allard and Rutenberg 2009; Zhou et al. 2017; Jung and Ha 2019). However, a full picture on the conditions to form a helix for an intrinsically straight filament is yet elusive. Particularly, the roles of different torques are not yet transparent. Moreover, it has been reported that under an applied force, the extension of a filament can subject to a sharp transition for either an intrinsically curved filament in 3D space or an intrinsically straight filament confined on a cylinder (Zhou et al. 2005, 2007; Zhou 2018; Allard and Rutenberg 2009; Zhou et al. 2017). Two relevant questions are then intrigue. The first one is that would torques also induce the same behavior, and the second is what kind of torque would have stronger effect for the transition? In this work, we find some algebraic static equations for a helix to obtain some exact results to answer these questions. We study the effects of two typical torques, one is parallel to the direction of a uniaxial applied force and another is coupled to the cross section of the filament. We show that BCs affect the results seriously. In 3D space, we find that the extension increases monotonically with increasing force so that there is not abrupt transition. However, for a confined filament with a fixed BC, the extension can subject to a first-order phase transition. We also present several phase diagrams to provide conditions to form a helix. Our findings suggest that under some conditions, an intrinsically straight semiflexible biopolymer can also act as a switch or sensor in some biological processes. Moreover, note that the fixed BC is analogous to prestressing a filament and in engineering, such as to build houses or bridges, it is commonly to use prestressed steel wires, the effect of a fixed BC may be instructive to engineering.

The paper is organized as follows. In the next section, we set up elastic models for an intrinsically straight filament. It follows a section on the mechanical property of a helix in 3D space. In Sect. 4, we focus on a helix confined on a cylinder. The section with conclusions and discussions completes the main text of the paper. Finally, we provide a appendix to discuss how to realize torques used in this work.

Models

Energy of a filament in 3D space

The configuration of a filament is analogous to the trajectory of a rigid plate so that we can use Euler angles \(\theta \), \(\phi \) and \(\psi \), as shown in Fig. 1, to describe it (Tanaka and Takahashi 1985; Benham 1989; Fain et al. 1997; Fain and Rudnick 1999; Panyukov and Rabin 2001; Zhou et al. 2005, 2007; Zhou 2018). It follows (Goldstein 2002)

$$\begin{aligned} \mathbf{t}_3& = (\sin \phi \sin \theta ,-\cos \phi \sin \theta ,\cos \theta ), \end{aligned}$$
(1)
$$\begin{aligned} \omega _1& = \sin \theta \sin \psi \ {\dot{\phi }}+\cos \psi \ {\dot{\theta }}, \end{aligned}$$
(2)
$$\begin{aligned} \omega _2& = \sin \theta \cos \psi \ {\dot{\phi }}-\sin \psi \ {\dot{\theta }}, \end{aligned}$$
(3)
$$\begin{aligned} \omega _3& = \cos \theta \ {\dot{\phi }}+{\dot{\psi }}. \end{aligned}$$
(4)

and \(\mathbf{r}(s)=(x, y, z)=\int _0^s\mathbf{t}_3(u)\mathrm{d}u\). The main advantage of using Euler angles is that we can find some algebraic static equations and so can obtain some exact results.

Fig. 1
figure 1

Definition of the Eulerian angles. \(\varvec{\xi }\) is the line of nodes which is a line perpendicular to tangent of the centerline and lies on the horizontal plane of the fixed frame. x, y and z are unit vectors on the axis of the fixed coordinate system

Full size image

In this work, we focus on two kinds of torque which are relatively easier to realize. Both torques are applied to \(s=L\). The first one is \(N_z\) which is along the direction of z-axis, and gives an energy density \(N_z({\dot{\phi }}+\cos \theta {{\dot{\psi }}})\), as derived in Appendix. The second one is \(N_3\) which is along the axis of filament, and provides an energy density \(N_3 \omega _3\), as derived in Appendix. For simplicity, we study their effects separately in this work, but it is easy to find more complicate results since we provide algebraic static equations.

In 3D space, the energy of an intrinsically straight and uniform filament with an intrinsic twist rate \(\omega _{0}\) can be written as

$$\begin{aligned} E& = \int _0^L{\mathcal {E}} \mathrm{d}s, \end{aligned}$$
(5)
$$\begin{aligned} {\mathcal {E}}& = {\mathcal {E}}_0 -F \cos \theta -N_z({\dot{\phi }}+\cos \theta {{\dot{\psi }}})-N_3\omega _3, \end{aligned}$$
(6)
$$\begin{aligned} {\mathcal {E}}_0& = {k_1\over 2}\omega _1^2 +{k_2\over 2}\omega _2^2+{k_3\over 2}\left( \omega _3-\omega _0\right) ^2, \end{aligned}$$
(7)

where \(k_1\) and \(k_2\) are bending rigidities, \(k_3\) is twisting rigidity, F is an uniaxial force along z-axis, L is the total contour length and is a constant; i.e., we consider an inextensible filament. F is also applied to \(s=L\). When \(k_1=k_2\), the filament is isotropic or has a circular cross section. When \(k_1=k_2\) and \(N_z=0\), it recovers the usual form of WLRC model (Fain et al. 1997; Marko and Siggia 1994; Bustamante et al. 1994; Marko and Siggia 1998).

Replacing \(N_3\) by \(N_3'=N_3+k_3\omega _0\), \({\mathcal {E}}\) becomes

$$\begin{aligned} {\mathcal {E}}={k_1\over 2}\omega _1^2 +{k_2\over 2}\omega _2^2+{k_3\over 2}\omega _3^2-F \cos \theta -N_z({\dot{\phi }}+\cos \theta {{\dot{\psi }}})-N_3'\omega _3+{k_3\over 2}\omega _0^2. \end{aligned}$$
(8)

Therefore, in fact \(\omega _{0}\) can be merged with \(N_3\), so that we will ignore \(\omega _0\) henceforth. Moreover, in numerical calculations and figures, we take \(k_1=1\) for simplicity.

Energy of a filament confined on a cylinder

Let the axis of the cylinder be along the z-axis, confining a filament on the surface of a cylinder of radius R applies a constraint on the coordinates so that \(x=R(1-\cos \phi )\) and \(y=-R\sin \phi \), as derived in Appendix. Comparing with Eq. (1), we obtain \({{\dot{\phi }}}=\sin \theta /R\) (Allard and Rutenberg 2009; Zhou et al. 2017). Applying a force, which is perpendicular to z-axis, at \(\mathbf{r}(L)\) results in a torque along the z-axis, and an energy density \(N_z {\dot{\phi }}\), as derived in Appendix. The energy for the filament confined on a cylinder is therefore

$$\begin{aligned} E& = \int _0^L{\mathcal {E}} \mathrm{d}s, \end{aligned}$$
(9)
$$\begin{aligned} {\mathcal {E}}& = {\mathcal {E}}_{0} -F \cos \theta -N_z \sin \theta /R-N_3 \omega _3 , \end{aligned}$$
(10)

where \(N_3\) is the same as that in Eq. (6), but \(N_z\) has a different meaning from that in Eq. (6). This is because \(N_z\) in Eq. (10) comes from a single force but to form the \(N_z\) in Eq. (6) requires a pair of forces, as we can see from Appendix. When \(N_3=0\), the model with free BC has been studied extensively (Allard and Rutenberg 2009; Zhou et al. 2017), but the role of \(N_3\) is yet unclear.

For simplicity, henceforth for a filament confined on a cylinder we will scale all lengths by R and the force by \(k_1/R^2\), which corresponds to letting \(R=1\) and \(k_1=1\) in E. The units of energy and torque are therefore \(k_1/R\).

Definition of a helix

In terms of Eulerian angles, the curvature \(\kappa \) and torsion \(\tau \) of a filament are (Zhou 2018)

$$\begin{aligned} \kappa& = \sqrt{{{\dot{\theta }}}^2+\sin ^2\theta {\dot{\phi }}^2}, \end{aligned}$$
(11)
$$\begin{aligned} \tau& = \cos \theta {\dot{\phi }}+{\sin \theta ({{\dot{\theta }}}\ddot{\phi }-{{\dot{\phi }}} \ \ddot{\theta })+\cos \theta {\dot{\phi }}{{\dot{\theta }}}^2\over \kappa ^2}. \end{aligned}$$
(12)

Note that in general \(\tau \ne \omega _3\) (Zhou 2018).

A general helix is defined as a curve in which \(\mathbf{t}_3\) makes a constant angle with a fixed direction. This condition is equivalent to having a s-independent \(\kappa /\tau \). In this work, we always apply a force along z-axis, so that the symmetry implies that \(\theta \) is s-independent. Therefore, we define the relative extension as \(z_r\equiv z(L)/L=\cos \theta \) for a helix. Without loss of generality, we also choose \(1\ge z_r\ge 0\). When both \(\kappa \) and \(\tau \) are s-independent, the helix is called a circular helix.

Static equations

In this work, we focus on GSCs. For a short rigid filament, the thermal effect is less important unless in some critical cases, since the rigidity tends to depress configurational fluctuation, so that at a first approximation we can ignore thermal fluctuation and find the GSCs by minimizing energy. Applying the standard variational technique results in the following static equations:

$$\begin{aligned} {\partial {\mathcal {E}}\over \partial \theta } -{d\over ds}{\partial {\mathcal {E}}\over \partial {{\dot{\theta }}}}={\partial {\mathcal {E}}\over \partial \phi } -{d\over ds}{\partial {\mathcal {E}}\over \partial {{\dot{\phi }}}}={\partial {\mathcal {E}}\over \partial \psi } -{d\over ds}{\partial {\mathcal {E}}\over \partial {{\dot{\psi }}}}=0, \end{aligned}$$
(13)

and boundary conditions (BCs) at \(s=0\) and \(s=L\)

$$\begin{aligned} {\partial {\mathcal {E}}\over \partial {{\dot{\theta }}}}\delta \theta ={\partial {\mathcal {E}}\over \partial {{\dot{\phi }}}}\delta \phi ={\partial {\mathcal {E}}\over \partial {{\dot{\psi }}}}\delta \psi =0. \end{aligned}$$
(14)

On a cylinder, we ignore the equations related to \(\phi \) and \({{\dot{\phi }}}\) in Eqs. (13)–(14).

Physically, for an arbitrary variable X, \(\delta X(x)=0\) means to fix X at x. Therefore, \(\delta \phi (L)=0\) corresponds to fix \(\phi (L)\) but \(\partial {\mathcal {E}}/\partial {{\dot{\phi }}}|_{s=L}=0\) corresponds to have a free \(\phi (L)\). In variational method, to fix X(0) and X(L) is equivalent to introduce an effective term \(-\alpha _X {\dot{X}}\) in energy density where \(\alpha _X\), such as \(C_\phi \) or \(C_\psi \) used hereafter, is a BC-dependent constant and is the generalized force required to fix X. In other words, to realize these constraints requires some complicate forces. Note that under a finite torque, to have a static configuration it is always necessary to fix \(\phi (0)\) and \(\psi (0)\) so that we always take \(\delta \phi (0)=\delta \psi (0)=0\). We also do not fix \(\theta (0)\) and \(\theta (L)\) in this work, since owing to the choice of F, a helix requires a s-independent \(\theta \) so it satisfies \(\partial {\mathcal {E}}/ \partial {{\dot{\theta }}}={{\dot{\theta }}}=0\) at both \(s=0\) and \(s=L\) automatically.

For a biopolymer, fixing \(\phi (L)\) or \(\psi (L)\) can be realized by binding two ends with some molecules. From Fig. 1, we know that to fix \(\phi (L)\) means to fix the line of nodes at \(s=L\), and the cross section of the filament can still rotate around \(\mathbf{t}_3\). On the other hand, fixing \(\psi (L)\) requires to specify a line on the cross section as the line of nodes, but the line of nodes can still rotate around z-axis.

When torques and BCs are given, a stable configuration requires \(g\equiv \mathrm{d}F/\mathrm{d}z_r > 0\) or extension increases with increasing force. Moreover, if g has more than one real zeros, there will be an abrupt change in \(z_r\) (Zhou et al. 2005, 2007; Zhou 2018; Zhou et al. 2017).

Mechanical property of a helix in 3D space

When \(k_1=k_2\)

In this case, Eq. (13) becomes

$$\begin{aligned}&[(k_3-k_1)\cos \theta {{\dot{\phi }}}^2 -(N_3-k_3{{\dot{\psi }}}) {{\dot{\phi }}}-N_z {{\dot{\psi }}}-F]\sin \theta +k_1\ddot{\theta }=0, \end{aligned}$$
(15)
$$\begin{aligned}&k_3z_r{{\dot{\psi }}}+(k_3\cos ^2\theta +k_1\sin ^2\theta ){{\dot{\phi }}}-N_z-N_3 \cos \theta =C_\phi , \end{aligned}$$
(16)
$$\begin{aligned}&k_3({{\dot{\psi }}} +\cos \theta {{\dot{\phi }}})-N_z \cos \theta -N_3=C_\psi , \end{aligned}$$
(17)

and BC for \(\theta \) is \({{\dot{\theta }}}=0\) at both \(s=0\) and \(s=L\).

From Eqs. (15)–(17), we know that \(\ddot{\theta }\), \({{\dot{\theta }}}\), \(\ddot{\phi }\) and \({{\dot{\phi }}}\) can all be expressed as functions of \(\theta \). Therefore, from Eqs. (11)–(12), requiring \(\kappa /\tau \) be s-independent results in a s-independent \(\theta \). It follows that \({{\dot{\phi }}}\) and \({{\dot{\psi }}}\) are also s-independent so we obtain some algebraic static equations.

With free BCs

In this case \(C_\phi =C_\psi =0\), so from Eqs. (15)–(17), it is straightforward to find that for a helix

$$\begin{aligned} F& = N_z [-k_1 N_3+(k_3-k_1)N_z z_r]/k_1k_3. \end{aligned}$$
(18)

Therefore, to form a helix, a finite \(N_z\) is a necessity but \(N_3\) can be zero and it is also irrelevant to stability. Moreover, F is a linear function of \(z_r\), i.e., the filament becomes a Hooke’s spring, and when \(k_3>k_1\), a helix is stable, but if \(k_1>k_3\), a helix is always unstable.

When \(\phi (L)\) is fixed

Now we consider the case with a fixed \(\phi (L)\) but a free \(\psi (L)\) so \(C_\psi =0\). In this case, \({{\dot{\phi }}}=[\phi (L)-\phi (0)]/L\) is determined by BC, and from Eqs. (15)–(17), we find

$$\begin{aligned} F& = -[N_z N_3+(N_z^2-2 k_3 N_z {{\dot{\phi }}} +k_1k_3{{\dot{\phi }}}^2)z_r]/k_3, \end{aligned}$$
(19)
$$\begin{aligned} g& = -(N_z^2-2 k_3 N_z {{\dot{\phi }}} +k_1k_3{{\dot{\phi }}}^2)/k_3. \end{aligned}$$
(20)

From Eqs. (19)–(20), we know that F is also a linear function of \(z_r\) and a large \(N_z\) or \({{\dot{\phi }}}\) makes a helix unstable. From Eq. (20), we can show exactly that when \(k_1>k_3\), a helix is always unstable since \(g<0\), and when \(k_3>k_1\), a helix is stable only when \(N^+>N_z>N^-\) with \(N^\pm =[k_3 \pm \sqrt{k_3(k_3-k_1)}]{{\dot{\phi }}}\). \(N_3\) is still irrelevant to stability. Comparing with the results for free BCs, fixing \(\phi (L)\) is not a proper way to form a helix since it can form a Hooke’s spring in a limited range of \(N_z\) only so it is uneasy to realize.

When \(\psi (L)\) is fixed

Next, we consider the case with a fixed \(\psi (L)\) but a free \(\phi (L)\) so \(C_\phi =0\). In this case, \({{\dot{\psi }}}=[\psi (L)-\psi (0)]/L\) is s-independent and from Eqs. (15)–(17), we obtain

$$\begin{aligned} {{\dot{\phi }}}& = [N_z+(N_3 -k_3{{\dot{\psi }}})z_r]/[k_1+(k_3-k_1)z_r^2]. \end{aligned}$$
(21)
$$\begin{aligned} F& = \left[ -k_1N_z[N_3-(k_3-k_1){{\dot{\psi }}} ]\right. \nonumber \\& \quad+[(k_3-k_1)N_z^2-k_1(N_3-k_3{{\dot{\psi }}})^2]z_r \nonumber \\& \quad+(k_3-k_1)N_z[N_3-(2k_1+k_3){{\dot{\psi }}} ]z_r^2\nonumber \\& \quad \left. -(k_3-k_1)^2N_z{{\dot{\psi }}} z_r^4\right] /\left[ k_1+(k_3-k_1)z_r^2\right] ^2. \end{aligned}$$
(22)

From Eq. (22), we find that when \(N_z=0\),

$$\begin{aligned} g=-{k_1 [k_1 + 3 (k_1 - k_3) z_r^2] (N_3 -k_3 {{\dot{\psi }}})^2\over [k_3 z_r^2 + k_1 (1 - z_r^2)]^3}. \end{aligned}$$
(23)

Therefore, when \(k_3<4k_1/3\), a helix is unstable but when \(k_3>4k_1/3\), a helix is stable if \(z_r>z_1\equiv \sqrt{k_1/(3k_3-3k_1)}\).

In contrast, when \(N_3=0\), from Eq. (22) we obtain

$$\begin{aligned} g& = N_z^2\{k_1[(k_1-k_3)+k_1k_3^2 \gamma ^2]- 6k_1k_3(k_1-k_3)\gamma z_r\nonumber \\& \quad +3(k_1 - k_3)[(k_1-k_3)+k_1k_3^2 \gamma ^2]z_r^2 \nonumber \\& \quad -2(k_1-k_3)^2k_3\gamma z_r^3\}[(k_1 - k_3) z_r^2-k_1]^{-3}, \end{aligned}$$
(24)

where \(\gamma ={{\dot{\psi }}}/N_z\).

From Eq. (24), we find that when \({{\dot{\psi }}}=0\), \(g=N_z^2(k_3-k_1)[k_1+3(k_1-k_3)z_r^2]/[k_1+(k_3-k_1)z_r^2]^3\) so that \(g<0\) when \(k_3<k_1\). In contrast, when \(k_3>k_1\), a helix is stable if \(z_r<z_1\). It follows that when \(k_3>4/3k_1\), a helix is unstable at large \(z_r\) since \(z_1<1\), but if \(k_3<4/3k_1\), a helix is stable since \(z_1>1\).

When \(N_3=0\) but \(\gamma \ne 0\), \(g=-k_1{{\dot{\psi }}}^2\le 0\) when \(k_3=k_1\) so that a helix is unstable. Furthermore, it is straightforward to show that \(dg/dz_r=0\), \(d^2g/dz_r^2<0\) and \(g< 0\) at \(z_r=z_0=k_3 \gamma /(k_3-k_1)\) when \(k_3<k_1\). In other words, \(g< 0\) and g reaches maximum at \(z_0\) when \(k_3<k_1\). Consequently, to form a helix, it requires \(k_3> k_1\).

Moreover, from Eq. (24), we find that when \(N_3=0\), the phase diagram, Fig. 2, for a helix can be divided into six regimes separated by \(\gamma _{0}^\pm \), \(\gamma _{1}^\pm \), \(k_3=k_1\) and \(k_3=4k_1\), with

$$\begin{aligned} \gamma _{0}^\pm& = \pm \sqrt{k_3-k_1}/(k_3\sqrt{k_1}), \end{aligned}$$
(25)
$$\begin{aligned} \gamma _{1}^\pm& = {(k_3-k_1)(k_3-4k_1)\pm k_3^{3/2}\sqrt{k_3-k_1} \over k_1k_3(4k_1-3k_3)}. \end{aligned}$$
(26)

\(g=0\) at \(z_r=0\) or \(z_r=1\) when \(\gamma =\gamma _{0}^\pm \) or \(\gamma =\gamma _{1}^\pm \), respectively. Consequently, when \(\gamma \) is far from \(\gamma =\gamma _{0}^\pm \), a helix with small \(z_r\) may be unstable; when \(\gamma \) is far from \(\gamma =\gamma _{1}^\pm \), a helix with large \(z_r\) may be unstable.

Fig. 2
figure 2

Phase diagram for a helix when \(k_1=k_2=1\) and \(N_3=0\). \(\gamma _0^-\) (red dashed), \(\gamma _0^+\) (green dotted), \(\gamma _1^-\) (black solid) and \(\gamma _1^+\) (blue dash dotted) are plotted

Full size image

In regime I, a helix is always stable since \(g>0\) and it occurs when \(4k_1>k_3>k_1\) and \(\gamma _1^+>\gamma >\gamma _0^-\). In other words, to form a helix at any \(z_r\), \(k_3\) and \(|\gamma |\) cannot be too large. A typical sample is shown as the black solid line in Fig. 3.

In regime II, \(g>0\) or a helix is stable when \(z_\mathrm{{II}}>z_r>0\) and it occurs when \(4k_1>k_3>k_1\) and \(\gamma _0^+>\gamma >\gamma _1^+\) or \(k_3>4k_1\) and \(\gamma _1^->\gamma >\gamma _0^-\). \(z_\mathrm{{II}}<1\) so that a helix of large \(z_r\) cannot exist. The value of \(z_\mathrm{{II}}\) is dependent on \(k_3\) and \(\gamma \). The green dashed line in Fig. 3 shows a typical sample in regime II.

Fig. 3
figure 3

\(z_r\) vs \(F'\equiv F/N_z^2\) when \(k_3=1.2\) and \(\gamma =-0.3\) (black solid); \(k_3=1.5\) and \(\gamma =0.1\) (green dashed); \(k_3=5\) and \(\gamma =-0.6\) (red dotted); \(k_3=9\) and \(\gamma =-0.4\) (blue dash dotted); \(k_3=12\) and \(\gamma =0.2\) (magenta short dashed). \(k_1=k_2=1\) and \(N_3=0\) in all cases

Full size image

In regime III, a helix is stable when \(1>z_r>z_\mathrm{{III}}>0\) and it occurs in three cases. The first case requires \(4/3k_1>k_3>k_1\) and \(\gamma _0^->\gamma >\gamma _1^-\) simultaneously; the second case needs \(4k_1>k_3>4/3k_1\) and \(\gamma _0^- >\gamma \) or \(\gamma >\gamma _1^-\); and the third case requires \(k_3>4k_1\) and \(\gamma <\gamma _1^+\) or \(\gamma >\gamma _0^+\). A helix of small \(z_r\) cannot exist in this regime. The red dotted line in Fig. 3 shows a typical sample in the regime. The value of \(z_\mathrm{{III}}\) is also dependent on \(k_3\) and \(\gamma \).

In regime IV, a helix is stable when \(z_\mathrm{{IV}}^1>z_r>z_\mathrm{{IV}}^0\) and it occurs when \(k_3>4k_1\) and \(\gamma _0^->\gamma >\gamma _1^+\). The blue dash dotted line in Fig. 3 shows a typical sample in regime IV. The values of \(z_\mathrm{{IV}}^0\) and \(z_\mathrm{{IV}}^1\) are also dependent on \(k_3\) and \(\gamma \).

In regime V, a helix is stable when \(z_r>z_\mathrm{{V}}^1\) or \(z_r<z_\mathrm{{V}}^0\), with \(z_\mathrm{{V}}^1>z_\mathrm{{V}}^0\) and it occurs when \(k_3>4k_1\) and \(\gamma _0^+>\gamma >\gamma _1^-\). The magenta short dashed line in Fig. 3 shows a typical sample in regime V. But note that in this regime, the force in \(z_r<z_\mathrm{{V}}^0\) is larger than that in \(z_r>z_\mathrm{{V}}^1\), which means that a larger force would result in a smaller \(z_r\), so that these two branches cannot be both in stable states, but one of which should be in a metastable state.

Finally, in regime VI, there is not helix and it occurs when \(4/3k_1>k_3>k_1\) and \(\gamma _1^- >\gamma \) or \(\gamma >\gamma _0^+\), or \(4k_1>k_3>k_1\) and \(\gamma _1^->\gamma >\gamma _0^+\). In other words, a negative or large \(\gamma \) but a small \(k_3\) does not favor a helix.

In summary, when \(k_1=k_2\), to have a stable helix requires \(k_3>k_1\). The relation between F and \(z_r\) is dependent on BCs. With free BC or fixing \(\phi (L)\), the helix is a Hooke’s spring and \(N_3\) is irrelevant. In other words, when \(\psi (L)\) is free, the twist around the cross section is decoupled from the bending of the centerline. Fixing \(\phi (L)\) also makes a helix stable only in a limited range of \(N_z\) so it is not a proper way to realize a helix. On the other hand, fixing \(\psi (L)\) results in a non-Hooke’s helix which may exist only in a certain range of \(z_r\) and it requires a complicate relation between \(\gamma \) and \(k_3\). In all cases, F is a monotonic function of \(z_r\) so that there is not any abrupt change in \(z_r\).

When \(k_1\ne k_2\)

When \(k_1\ne k_2\), there is not a direct way to show exactly that \(\theta \) must be s-independent for a helix. However, note that both F and \(N_z\) are along z-axis and the main role of \(N_3\) is to distort the cross section so should affect less on shape of the centerline; from symmetry, it is reasonable to expect that the axis of a helix is also along z-axis so that \(\theta \) is also s-independent. Consequently, Eq. (13) is reduced into

$$\begin{aligned}&(k_1-k_2) \sin (2\psi )\ddot{\phi }-2N_z{{\dot{\psi }}} -2 F\nonumber \\& \quad -2[N_3-(k_3+(k_1-k_2)\cos (2\psi )){{\dot{\psi }}}] {{\dot{\phi }}}\nonumber \\& \quad +z_r [2 k_3-k_1-k_2+(k_1-k_2)\cos (2\psi )]{{\dot{\phi }}}^2=0, \end{aligned}$$
(27)
$$\begin{aligned}&[k_3 z_r^2+k_2 (1-z_r^2)\cos ^2\psi +k_1 (1-zr^2)\sin ^2\psi ]{{\dot{\phi }}} \nonumber \\& \quad +k_3z_r{{\dot{\psi }}}-N_z-N_3z_r=C_\phi , \end{aligned}$$
(28)
$$\begin{aligned}&2k_3(z_r\ddot{\phi }+ \ddot{\psi }) - (k_1-k_2)(1-z_r^2)\sin (2\psi ){{\dot{\phi }}}^2=0. \end{aligned}$$
(29)

with BCs

$$\begin{aligned} \sin (2\psi ){{\dot{\phi }}} \delta \theta& = 0, \end{aligned}$$
(30)
$$\begin{aligned} \left[ k_3 ({{\dot{\psi }}}+z_r {{\dot{\phi }}}) -N_3- N_z z_r \right] \delta \psi& = 0. \end{aligned}$$
(31)

Equations (27)–(29) are second-order nonlinear differential equations of \(\phi \) and \(\psi \). It is straightforward to show that in general these equations are incompatible if \(\ddot{\phi }\ne 0\) or \(\ddot{\psi }\ne 0\). Therefore, to form a helix, it requires \(\ddot{\phi }=\ddot{\psi }= 0\) and it follows \(\psi =0\) or \(\psi =\pi /2\) from Eq. (29). Moreover, we can show that taking \(\psi =\pi /2\) is equivalent to taking \(\psi =0\) and exchanging \(k_1\) and \(k_2\) in Eqs. (27)–(28). In other words, for a helix we only need to consider the case with \(\psi =0\).

It is also straightforward to show that when both \(\phi (L)\) and \(\psi (L)\) are free, Eq. (31) is inconsistent with Eq. (28) so that there is not stable helix.

With free \(\phi (L)\)

With free \(\phi (L)\) so \(C_\phi =0\), taking \(\psi =\psi (L)=0\), from Eqs. (27)–(31) we obtain

$$\begin{aligned} {{\dot{\phi }}}& = (N_z+N_3z_r)/[k_2+(k_3-k_2)z_r^2], \end{aligned}$$
(32)
$$\begin{aligned} F& = {(N_z + N_3 z_r) [(k_3-k_2) N_z z_r - k_2 N_3]\over [k_2+(k_3-k_2)z_r^2]^2}. \end{aligned}$$
(33)

When \(N_z=0\),

$$\begin{aligned} g={k_2 N_3^2 [-k_2 + 3 (k_3-k_2) z_r^2]\over [k_3z_r^2 + k_2 (1 - zr^2)]^3}. \end{aligned}$$
(34)

Meanwhile, when \(N_3=0\),

$$\begin{aligned} g={N_z^2 (k_3 - k_2) (k_2+3(k_2-k_3) z_r^2)\over [k_3z_r^2 + k_2 (1 - zr^2)]^3}. \end{aligned}$$
(35)

It follows that when \(k_3<k_2\), \(g<0\) so that a helix is unstable in either \(N_3=0\) or \(N_z=0\). A helix can be stable only when \(1\ge z_r>\sqrt{k_2/3 (k_3-k_2) }\) and \(k_3>4k_2/3\) if \(N_z=0\), or when \(z_r< \sqrt{k_2/3 (k_3-k_2) }\) if \(N_3=0\). Therefore, \(N_z\) plays an opposite role to \(N_3\) in this case. Moreover, g has only one zero for \(1>z_r>0\) so that F is a smooth function of \(z_r\).

When \(\phi (L)\) is fixed

On the other hand, when \(C_\phi \ne 0\) but \(\psi =0\), \({{\dot{\phi }}}\) is s-independent and Eqs. (27)–(31) result in

$$\begin{aligned} {{\dot{\phi }}}& = (N_3+N_zz_r)/k_3z_r, \end{aligned}$$
(36)
$$\begin{aligned} F& = (N_3+N_zz_r)[(k_3-k_2)N_z z_r-k_2N_3]/k_3^2z_r. \end{aligned}$$
(37)

When \(N_z=0\), \(F=-k_2N_3^3/k_3^2 z_r<0\) is a compressive force and a helix is stable at arbitrary \(k_2\) and \(k_3\).

When \(N_3=0\), \(F=(k_3-k_2)N_z^2z_r/k_3^2\) so that to form a helix requires \(k_3>k_2\).

In summary, with free BCs, anisotropy prohibits a helix. With fixing \(\phi (L)\), the filament becomes a non-Hooke’s spring which may exist only in a certain range of \(z_r\). In all cases, F is a monotonic function of \(z_r\) so that there is not abrupt change in \(z_r\).

Mechanical properties of a helix confined on a cylinder

On a cylinder or inside a cell, binding two ends of a biopolymer is more feasible than that in 3D space, so that how to form a helix on a cylinder should be a more intrigue issue.

When \(k_1=k_2\)

From Eq. (13), we obtain the following static equations

$$\begin{aligned} {{\dot{\psi }}}& = N_3/k_3-\sin \theta \cos \theta +C_\psi , \end{aligned}$$
(38)
$$\begin{aligned} 4k_1 \ddot{\theta }& = 4 k_3 \cos (2\theta ){{\dot{\psi }}} +(k_3-k_1)\sin (4\theta )+2k_1 \sin (2\theta )\nonumber \\& \quad + \, 4F\sin \theta -4N_z\cos \theta -4 N_3 \cos (2\theta ). \end{aligned}$$
(39)

Similar to the 3D case, from Eqs. (38)–(39), we find that \(\ddot{\theta }\), \({{\dot{\theta }}}\), \(\ddot{\psi }\) and \({{\dot{\psi }}}\) can be expressed as functions of \(\theta \), so that requiring \(\kappa /\tau \) be s-independent results in a s-independent \(\theta \). It follows that \({{\dot{\psi }}}\) is also s-independent so we obtain two algebraic equations again.

When \(\psi (L)\) is free, \(C_\psi =0\) and

$$\begin{aligned} k_1 \ddot{\theta }& = 2 k_1 \sin ^3 \theta \cos \theta +F\sin \theta -N_z\cos \theta . \end{aligned}$$
(40)

Equation (40) indicates that \(\theta \) is independent of \(k_3\) and \(N_3\), or \(k_3\) and \(N_3\) do not affect the shape of the centerline so they can be ignored. \(N_3\) affects only the twist around the centerline. It suggests that to obtain a helix, applying a \(N_z\) is an easier and more efficient way. The relevant system with a moderate length has been studied extensively (Allard and Rutenberg 2009; Zhou et al. 2017), and the main conclusions are (Zhou et al. 2017): Thermal effect and excluded volume interaction are less important; a uniaxial force results in a helix of large \(z_r\) only, and at a critical F the filament can collapse from a helix to a non-helix; \(N_z\) alone is enough to stabilize a helix to a very small \(z_r\), and confinements from both ends reduce \(N_z\) effectively. Finally, there is not abrupt transition in extension between helices of different \(z_r\).

Owing to thermal fluctuation, at a finite temperature BCs affect little a long filament. Consequently, at a finite temperature, \(k_3\) and \(N_3\) can be ignored for a long filament confined on a cylinder.

However, a semiflexible biopolymer may be short so that it is unreasonable to ignore BCs or \(k_3\) and \(N_3\). For instance, a MreB helix inside a cylindrical bacteria has only few turns (Jones et al. 2001; Iwai et al. 2002; Daniel and Errington 2003; Gitai et al. 2004, 2005; Kruse et al. 2005; Carballido-López 2006; Vats and Rothfield 2007). Therefore, we still need to consider the case with a fixed \(\psi (L)\) and find it yields quite different results. In this case, we have

$$\begin{aligned} F& = [N_z z_r-N_3' (1-2z_r^2)]/\sqrt{1-z_r^2} \nonumber \\& \quad +[k_3-2k_1+2(k_1-k_3)z_r^2]z_r. \end{aligned}$$
(41)
$$\begin{aligned} 2{\mathcal {E}}& = 1+(N_3'^2-N_3^2)/k_3 +2 z_r^2-3 z_r^4 -{2( N_z + N_3' z_r^3) \over \sqrt{1-z_r^2}}. \end{aligned}$$
(42)

\(N_3'\equiv N_3-k_3{{\dot{\psi }}}\) is an effective torque or fixing \({{\dot{\psi }}}\) plays a similar role as applying a finite \(N_3\). But note that \(N_3\) gives an additional constant in \({\mathcal {E}}\).

Equation (41) suggests that \(k_3\) and \(N_3'\) are important for a helix, and a positive \(N_z\) or \(N_3'\) favors a helix. Moreover, when \(z_r=0\), \(g =k_3+N_z-2k_1\) so that when \(N_z+k_3<2k_1\), a helix with a small \(z_r\) is unstable. When \(z_r\rightarrow 1\), \(g \rightarrow (N_3'+N_z)/\sqrt{1-z_r^2}\) so that a helix with a large \(z_r\) is stable if \(N_3'+N_z>0\).

When \(N_z=0\),

$$\begin{aligned} g=k_3 - 2 k_1 + 6 (k_1 - k_3) z_r^2+{N_3' (3-2z_r^2)z_r\over (1 - z_r^2)^{3/2}}. \end{aligned}$$
(43)

Equation (43) results in \(g<0\) when \(k_1>k_3>0.8k_1\) and \(N_3'<0\). When \(z_r=0\), \(g=k_3-2k_1\) so that it requires \(k_3>2k_1\) to form a helix at small \(z_r\). In another limit, \(z_r\sim 1\), the sign of g is determined by \(N_3'\) and it requires \(N_3'>0\) to have a helix at large \(z_r\). When \(N_3'=0\), \(g>0\) if \(z_r<\sqrt{(k_3-2k_1)/(6k_3-6k_1)}\).

Moreover, when \(N_z=0\), numerical calculations reveal that g can have not any real zero, have one or two or three real zero(s), and different number of zeros results in different behavior. Consequently, the phase diagram for a helix can be divided into six regimes, as shown in Fig. 4.

Fig. 4
figure 4

Phase diagram for a helix when \(k_1=k_2=1\) and \(N_z=0\). The red dashed line is approximately \(N_3'=-0.2831k_1+0.4022 k_3\). The black solid oblique line is approximately \(N_3'=-0.2286k_1+0.499k_3\). The vertical black line is given by \(k_3=2\)

Full size image

In regime I, it has always \(g>0\) at any \(z_r\) so that a helix is always stable. The regime is bounded by \(k_3>2\) and the black solid oblique line in Fig. 4. A typical sample is shown as the black solid line in Fig. 5.

In regime II, \(g> 0\) when \(z_\mathrm{{II}}> z_r\) so that a helix is stable when \(z_\mathrm{{II}}>z_r>0\) and it occurs when \(k_3>2k_1\) and \(N_3'<0\). The larger the \(N_3'\), the smaller the \(z_\mathrm{{II}}\). The magenta short dashed line in Fig. 5 shows a typical sample in the regime. Note that \(z_\mathrm{{II}}\) here is different from that in Sect. 3, and the same hints for other special values of \(z_r\) in the following texts.

In regime III, \(g> 0\) when \(z_r> z_\mathrm{{III}} \) so that a helix is stable when \(z_r>z_\mathrm{{III}}\). This regime is bounded by \(N_3'>0\), \(k_3>0\), the green dashed line, the black solid oblique line and the vertical black line in Fig. 4. The larger the \(N_3'\), the smaller the \(z_\mathrm{{III}}\). The green dashed line in Fig. 5 shows a typical sample in the regime.

In regime IV, \(g\le 0\) when \(z_r\le z_\mathrm{{IV}}^0\) or \(z_r\ge z_\mathrm{{IV}}^1\) and \(z_\mathrm{{IV}}^1>z_\mathrm{{IV}}^0\). It follows that a helix is stable when \(z_\mathrm{{IV}}^1>z_r>z_\mathrm{{IV}}^0\) and the regime is bounded by \(N_3'<0\) and the red dashed line in Fig. 4. The larger the \(N_3'\), the smaller the \(z_\mathrm{{IV}}^0\) but the larger the \(z_\mathrm{{IV}}^1\). A typical sample is shown as the blue dotted line in Fig. 5.

In regimes I–IV, \(z_r\) increases monotonically with increasing F so that there is not sharp transition in \(z_r\).

Fig. 5
figure 5

\(z_r\) vs F for a helix when \(k_3 =2\) and \(N_3'=1\) (black solid); \(k_3 =2.5\) and \(N_3'=-0.1\) (magenta short dashed); \(k_3 =1.5\) and \(N_3'=0.5\) (green dashed); \(k_3 =0.5\) and \(N_3'=-0.02\) (blue dotted). \(N_z=0\) and \(k_1=k_2=1\) in all cases

Full size image

Regime V is special because in the regime \(z_r\) shows a shape transition at some F. In this regime, \(g\le 0\) when \(z_\mathrm{V}^2\ge z_r\ge z_\mathrm{V}^1\) if \(k_3>2k_1\) so g has two zeros, or \(g\le 0\) when \(z_\mathrm{V}^2\ge z_r\ge z_\mathrm{V}^1\) or \(z_r\le z_\mathrm{V}^0\) if \(k_3<2k_1\) so g has three zeros. The larger the \(N_3'\), the larger the \(z_{\mathrm{V}}^1\) but the smaller the \(z_{\mathrm{V}}^2\). In other words, larger \(N_3'\) makes the transition sharper or more sensitive to force. The regime is bounded by \(N_3'>0\), the green dashed line and the black solid oblique line as shown in Fig. 4. Figure 6 shows a typical sample in the regime. Since g has two or three zeros, \(z_r\) is a triple-valued function of F when \(z_\mathrm{V}^2> z_r> z_\mathrm{V}^1\), as shown in Fig. 6. In the regime, \({\mathcal {E}}-F\) is self-crossed at \(z_r=z_\mathrm{E}\), and the crossover point gives the lowest energy under a given F. Therefore, in a quasi-static process, \(z_r\) will jump suddenly at crossover point of \({{\mathcal {E}}}\). Moreover, tips in both \(z_r-F\) and \({\mathcal {E}}-F\) curves define two metastable regimes, one is from \(z_{\mathrm{V}}^1\) to \(z_\mathrm{E}\) and the other is from \(z_\mathrm{E}\) to \(z_{\mathrm{V}}^2\). It means that in practice the discontinuous change in \(z_r\) will be more likely to occur at \(z_{\mathrm{V}}^1\) with increasing |F|, or occur at \(z_{\mathrm{V}}^2\) with decreasing |F|. The hysteresis indicates that the phase transition is first order. These behaviors are similar to those reported in Refs. (Zhou et al. 2005, 2007; Zhou 2018). Note that the transition occurs between two helices of different \(z_r\), and it is different from a system with free BC (Zhou et al. 2017).

Fig. 6
figure 6

\(z_r\) (solid) and \({\mathcal {E}}\) (dash dotted) vs F when \(N_z=0\), \(k_1=k_2=1\), \(k_3 =3.4\) and \(N_3'=1.28\). Two vertical dashed straight lines denote zeros of g and give \(z_\mathrm{V}^1\) and \(z_\mathrm{V}^2\). The vertical solid straight lines denote the crossover point of \({{\mathcal {E}}}\). \(N_3=0\) in \({{\mathcal {E}}}\)

Full size image

Finally, in regime VI, it has always \(g<0\) so that there is not helix. The regime is bounded by \(N_3'<0\), \(k_3<2k_1\) and the red dashed line in Fig. 4.

On the other hand, when \(N_3'=0\),

$$\begin{aligned} g=k_3 - 2 k_1 + 6 (k_1 - k_3) z_r^2+N_z/(1 - z_r^2)^{3/2}. \end{aligned}$$
(44)

In this case, when \(N_z=0\), g has a real zero in \(1>z_r>0\) when \(2k_1>k_3>8k_1/7\). When \(z_r= 0\), \(g= N_z+k_3-2k_1\) so that it requires \(N_z+k_3>2k_1\) to have a helix with \(z_r\sim 0\); when \(z_r\sim 1\), \(g\sim N_z/(1 - z_r^2)^{3/2}\) so that it requires \(N_z>0\) to form a helix with \(z_r\sim 1\).

The phase diagram when \(N_3'=0\) is shown in Fig. 7. The phase diagram can be divided into five regimes separated by \(N_z+k_3=2k_1\), \(N_z=0\) and the solid line in Fig. 7.

In regime I, \(N_z+k_3<2k_1\) and \(N_z>0\), so that a helix is unstable when \(z_r\sim 0\). In regime II, \(N_z+k_3<2k_1\) and \(N_z<0\), so that a helix is unstable in either \(z_r\sim 0\) or \(z_r\sim 1\). In regime III, \(N_z+k_3>2k_1\) and \(N_z<0\), so that a helix is unstable if \(z_r\sim 1\). In regimes \(I--III\), F is a smooth function of \(z_r\) or there is not sharp transition in \(z_r\).

Regime IV is bounded by \(N_z+k_3=2k_1\), \(N_z=0\) and the solid line in Fig. 7. This regime is also special because g has two real zeros. Consequently, a helix with either a small \(z_r\) or large \(z_r\) is stable, but there exists a critical regime for \(z_r\) in which \(z_r\) has a first-order transition with varying F, similar to that shown in Fig. 6. Note that the range of this regime is similar to the regime V when \(N_z=0\), and it indicates that the sharp transition can still occur when both \(N_z\ne 0\) and \(N_3'\ne 0\).

Finally, the regime V is bounded by \(N_z+k_3=2k_1\) and the solid line. In this regime, a helix is always stable and F is also a smooth function of \(z_r\).

Fig. 7
figure 7

Phase diagram for helix when \(N_3'=0\) and \(k_1=k_2=1\). The dash dotted line is given by \(N_z+k_3=2k_1\). The solid line is approximately \(N_z=-\,0.3098k_1+0.7045k_3\)

Full size image

Moreover, when \(N_3'=0\), we find that with proper parameters and under a given F, \(N_z\) can induce a sharp change in \(z_r\) for a helix. A typical example is shown in Fig. 8. From Fig. 8, we can see that beginning from a moderate \(z_r\), \(z_r\) is a triple-valued function of \(N_z\) and the \({\mathcal {E}}-N_z\) curve is self-crossed and the crossover point gives the lowest energy under a given \(N_z\). It means that in the triple-valued regime, \(z_r\) will subject to a first-order transition, similar to that reported in Refs. (Zhou et al. 2007; Zhou 2018). Again, the transition occurs between two helices of different \(z_r\) but there is not such a transition for the system with a free \(\psi (L)\) (Zhou et al. 2017).

Fig. 8
figure 8

\(z_r\) (solid) and \({\mathcal {E}}'={\mathcal {E}}+1.5\) (dash dotted) vs \(N_z\) when \(N_3'=0\), \(k_1=k_2=1\), \(k_3 =2.8\) and \(F=0.5\). \(N_3=0\) in \({{\mathcal {E}}}\). Two vertical dashed straight lines denote where \(G\equiv dN/dz_r=0\), and the vertical solid straight lines denote the crossover point of \({{\mathcal {E}}}\)

Full size image

In summary, when \(k_1=k_2\) and with a free \(\psi (L)\), \(k_3\) and \(N_3\) are irrelevant to a helix. In contrast, when \(\psi (L)\) is fixed, \(k_3\) and \(N_3\) are significant and result in rich phenomena owing to the coupling between bending and twisting. Particularly, a helix is in general a non-Hooke’s spring, and its extension can subject to a first-order transition. \(N_3\) and \(N_z\) play similar roles for the transition, and a large \(k_3\) and positive torques favor such a transition.

When \(k_1\ne k_2\)

In this case, in the same reason as that in subsection 3.2, taking a s-independent \(\theta \), the static equations become

$$\begin{aligned}&[k_3(1-2 z_r^2)+(k_1-k_2) (1-z_r^2) \cos (2\psi )]{{\dot{\psi }}} \nonumber \\& \quad -2(k_1\sin ^2\psi +k_2 \cos ^2\psi ) z_r(1-z_r^2)^{3/2}\nonumber \\& \quad +k_3z_r(1-2z_r^2)\sqrt{1-z_r^2} -N_3 (1-2z_r^2) +N_z z_r\nonumber \\& \quad -F \sqrt{1-z_r^2}=0, \end{aligned}$$
(45)
$$\begin{aligned}&2k_3 \ddot{\psi }- (k_1-k_2)(1-z_r^2)^2\sin (2\psi )=0. \end{aligned}$$
(46)

with BCs

$$\begin{aligned} \sin (2\psi )=0, \ (k_3 {{\dot{\psi }}} -N_3+k_3z_r\sqrt{1-z_r^2})\delta \psi =0. \end{aligned}$$
(47)

Equation (45) is a first-order nonlinear differential equation in \(\psi \), and Eq. (46) is a second-order nonlinear differential equation of \(\psi \). It is straightforward to show that in general these two equations are incompatible when \(k_1\ne k_2\) and \({{\dot{\psi }}}\ne 0\). Therefore, to form a helix, from Eq. (46) we know that it has \(\psi =0\) or \(\psi =\pi /2\).

Moreover, from Eq. (45), if \(\psi =0\) and \(\psi (L)\) is free, we obtain

$$\begin{aligned} N_3& = k_3z_r\sqrt{1-z_r^2}, \end{aligned}$$
(48)
$$\begin{aligned} F& = -2 k_2 z_r (1-z_r^2)+N_zz_r/\sqrt{1-z_r^2}. \end{aligned}$$
(49)

Eqs. (48)–(49) are the same as those when \(k_1=k_2\), \(\psi =0\) and \(C_\psi =0\), i.e., Eqs. (38) and (40) with \(\ddot{\theta }=0\), \(\psi =0\) and \(C_\psi =0\). In other words, the behavior of the helix is the same as that with \(N_3=0\), or anisotropy does not provide any new result.

Conclusions and discussions

In summary, we study the effects of two typical applied torques on the mechanical property of the helical configuration for an intrinsically straight filament in either three-dimensional space or on a cylinder. We obtain some algebraic static equations for a helix and find that the BCs affect the results seriously.

In 3D case, we find that for an isotropic filament, to have a stable helix requires that the twisting rigidity is larger than the bending rigidity. With free BCs or with a fixed \(\phi (L)\), the filament becomes a Hooke’s spring and a finite \(N_z\) is necessary but \(N_3\) is irrelevant or the twisting is decoupled from bending. Since at a finite temperature and for a long filament, the thermal fluctuation will reduce the effects of BCs, we can conclude that \(N_3\) is helpless in this case. On the other hand, fixing \(\psi (L)\) results in a non-Hooke’s spring which may exist only in a certain range of \(z_r\) and demands a complicate relation between \(\gamma \) and \(k_3\). Moreover, with free BCs, anisotropy prohibits a helix, but with a fixed BC, the anisotropic filament becomes a non-Hooke’s spring. In all cases, force is a monotonic function of extension so that there is not any abrupt change in extension for a 3D helix.

On the other hand, for an isotropic filament confined on a cylinder and when \(\psi (L)\) is free, again \(N_3\) is irrelevant in forming a helix. Consequently, at a finite temperature, \(k_3\) and \(N_3\) can also be ignored for a long filament. However, fixing \(\psi (L)\) makes \(k_3\) and \(N_3\) crucial and results in rich behaviors. Particularly, in general the filament becomes a non-Hooke’s spring, and with a large twisting rigidity and a large torque the extension of a helix can subject to a first-order transition. It is reasonable to expect that the similar phenomena will also occur in some other constrained systems. The transition occurs between two helices of different \(z_r\) so it may be relatively easier to be identified. For a biopolymer, fixing \(\psi (L)\) can be realized by binding the end with some molecules, so that this finding is significant for a short rigid biopolymer since such a biopolymer can act as a switch or sensor in some biological processes.

Our findings support the conclusion that ‘closed cylindrical confinement and chain stiffness are key factors for helical organization’ of a biopolymers (Jung and Ha 2019). Our results also offer some insights into the conditions to form a helix for some semiflexible biopolymers. For instance, the conclusion that in free space to form a helix requires \(k_3>k_1\) suggests that a double-stranded DNA (dsDNA) may form a helix but a MreB molecule cannot, since \(k_1/k_\mathrm{B}T \approx 52\pm 2\) nm and \(k_3/k_\mathrm{B}T \approx 75\pm 25\) nm for a dsDNA, but \(k_1/k_\mathrm{B}T \approx 3.79 \times 10^6\) nm and \(k_3/k_\mathrm{B}T \approx 2.54 \times 10^6\) nm for a MreB molecule, where \(k_\mathrm{B}\) is the Boltzmann constant and T is the temperature. However, inside a cylinder a MreB molecule can form a helix since its large \(k_3\) favors a helix. Our findings may also be instructive to engineering since to fix Euler angles is analogous to prestress a filament and to build houses or bridges, it is commonly to use prestressed steel wires.

In this work, we do not consider the thermal and excluded volume effects. However, for an isotropic filament confined on a cylinder and with a free \(\psi (L)\), it has been found that the thermal effect and excluded volume interaction play little role for a helix of moderate length (Zhou et al. 2017) and we expect that the same conclusion holds for the filament with a fixed \(\psi (L)\). Moreover, for a filament confined on a cylinder, we do not consider the confinement from both ends of the cylinder in this work. It has been reported that such a confinement is crucial for a helix under a finite \(N_z\) (Zhou et al. 2017), so that the effect of the confinement is intrigue when \(N_3\) is finite and it deserves a further investigation.