# Reshaping and Enzymatic Activity allow Viruses to move through the Mucus

Falko Ziebert, Kenan G. Dokonon, Igor Kulić

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Falko Ziebert, Kenan G. Dokonon, Igor Kulić. Reshaping and Enzymatic Activity allow Viruses to move through the Mucus. 2023. hal-04299177

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Reshaping and Enzymatic Activity allow Viruses to move through the Mucus

Falko Ziebert,¹, * Kenan G. Dokonon,² and Igor M. Kulić³, †

¹Institute for Theoretical Physics, Heidelberg University,
Philosophenweg 19, 69120 Heidelberg, Germany and
BioQuant, Heidelberg University, Im Neuenheimer Feld 267, 69120 Heidelberg, Germany

²Department of Chemical Engineering and Biotechnology,
University of Cambridge, Cambridge CB3 0AS, UK.

³Institut Charles Sadron UPR22-CNRS, 67034 Strasbourg, France and
Institute Theory of Polymers, Leibniz-Institute of Polymer Research, D-01069 Dresden, Germany

Filamentous viruses like influenza and torovirus often display systematic bends and arcs of mysterious physical origin. We propose that such viruses undergo an instability from a cylindrically symmetric to a toroidally curved state. This “toro-elastic” state emerges via a spontaneous symmetry breaking under prestress, induced via short range spike protein interactions and magnified by the filament’s surface topography. Once surface stresses become sufficiently large, the filament buckles and the toroidal, curved state constitutes a soft mode that can propagate through the filament’s material frame around a “mexican-hat” potential. In the mucus of our airways, glycan chains are omnipresent that influenza’s spike proteins can bind to and cut. We show that when coupled to such a non-equilibrium chemical reaction, the curved toro-elastic state can attain a spontaneous rotation for sufficiently strong enzymatic activity, leading to a whole body reshaping propulsion similar to – but different from – eukaryotic flagella and spirochetes.

Thin fibers are a common structural element in biology, from biofilaments to the body shapes of viruses and bacteria. Notably, biological filaments like the bacterial flagellar hook [1, 2], possibly microtubules [3, 4], and intermediate filaments [5] exhibit a highly unusual, common motif: their straight ground state can become unstable, reshaping them into polymorphic toroids and superhelices. Not less surprisingly, when driven out of equilibrium, stimuli-responsive but otherwise straight fibers can acquire a dissipative toroidal steady state of spontaneous rotation [6, 7]. Visually inspecting the shapes of filamentous viruses like Influenza C (IV-C) [8, 9] or Torovirus [10], see Fig. 1, the natural question arises if viral envelopes could possess a similar symmetry-broken, toroidal ground state. And as viruses like influenza A (IV-A) and C [11–13], as well as paramyxovirus [14] have been shown to be able to roll on surfaces due to their spike proteins’ catalytic activity, can this toroidal state be driven catalytically and used for virus propulsion?

In the following we consider a plausible model for the emergence of what we call the toro-elastic state of the viral envelope. It relies on surface switchability at the interfaces of spike proteins and a mismatch-stress with layers further inside the virus. The switch corresponds to the making-and-breaking of any reversible physical bond – including hydrogen bonds, screened-electrostatic, van der Waals bonds – as well as protein-tail-mediated interactions. Notably, recently resolved surface structures of influenza C [15] have shown extensive spike-spike contacts, establishing a hexagonal lattice on the envelope. The natural cylindrical curvature of this spike-spike network is further modulated by the internal transmembrane and nucleoprotein matrix proteins, indicating some form of core-shell strain mismatch for influenza C.

After demonstrating the mechanism giving rise to the

toro-elastic state, we explain how its continuous ground state looks much like a built-in “wheel within the material” [7], mirroring the “universal joint” model of the bacterial flagellar hook proposed by Namba and coworkers [2]. Finally, we study the effect of subtle dynamic force imbalances, generated by polymeric ligands being bound and cut on the viral envelope. This situation reflects what influenza is confronted within the mucus of our airways, which is a highly viscous environment containing polymers with sugary ends (glycans) the virus interacts with. We show that a broken detailed-balance via enzymatic cutting can force the toroidally curved state to spontaneously rotate once the enzyme surpasses a certain minimal activity threshold. This dynamic shape ro-

<figure><img src="image_3.png" /><figcaption>FIG. 1. Emergent toroidal shapes of various biofilaments: a) torovirus [18]; b) the bacterial flagellar hook [1]; c) Influenza C: long filamentous viruses show both straight and oscillatory shapes [9]; inset: short, curved filamentous IV-C actively rolling on glycan coated surfaces [12].</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>FIG. 2. Geometry of the virus and the toroidal mode: (a) spike proteins are arranged on a typically staggered column lattice with spacing $l_0$ and effective radius $R$ (including the spikes). (b) Curving the lattice by curvature $\kappa$ allows some spikes to interact at the inside of the bend. (c) Which particular spikes can interact depends on their orientation angle $\theta$ with respect to the angle $\phi$ of the curvature vector $\vec{\kappa}$.</figcaption></figure>

tation mode should allow the virus to move at small but noticeable speed, and could explain why catalytic activity increases mucus penetration so strongly for influenza [16, 17].

**Toro-elastic state.** Consider the virus surface covered with radial spikes – in case of influenza C called Hemagglutinin-esterase-fusion protein (HEF) – with initial spacing $l_0$ (along the filament axis, see Fig. 2a). If the virus curves, variations of strains $\epsilon$ and thus distances between the spikes are amplified proportional to the effective radius of the cylindrical virus, $R$, and the centerline curvature $\kappa$ like $l(\kappa) \approx \epsilon l_0 = \kappa R l_0$. In addition, there is a geometric “shortcut factor” for finite deformations: For spiky surface elements that interact along the shortest spatial distance, this distance is given by the secant line (see SI for details).

Introducing the vectorial curvature perpendicular to the axis $\vec{\kappa} = (\kappa_x, \kappa_y) = \kappa(\cos\phi, \sin\phi)$ with orientation angle $\phi$ and magnitude $\kappa$, and parameterizing the material-fixed spike positions with angle $\theta$ (with $\theta = \phi$ corresponding to the orientation of the curvature vector, i.e. towards the bend), the spacing between surface contacts is given by, see Fig. 2b,c and SI,

$$l(\kappa, \theta - \phi) = l_0 \frac{\sin(\kappa l_0 / 2)}{\kappa l_0 / 2} (1 - R\kappa \cos(\theta - \phi)). \quad (1)$$

Here the first term is the shortcut factor described above and the second shows the dependence on the angle differences. At this point we can fix $\vec{\kappa}$ to point in the x-direction, setting $\phi = 0$.

Every spike can interact with its axial neighboring spikes, in case they are within range, letting them gain a binding free energy (per unit length), $v(l)$. On the flip side, binding along its long axis implies a curving of the cylindrical virus, which costs elastic bending energy. Modeling the latter contribution as a harmonic term with bending stiffness $B$, the total energy per cross section is hence given by

$$E = E_{\text{bend}} + E_{\text{bind}} = \frac{1}{2} B \kappa^2 + \int_{0}^{2\pi} v(l(\kappa, \theta)) d\theta. \quad (2)$$

<figure><img src="image_5.png" /><figcaption>FIG. 3. (a) A spike interaction potential with a minimum at a finite distance $\sigma$. (b) The straight ground state can become unstable through spike-spike interactions and give rise to a Mexican hat-potential and a “toro-elastic” mode: in angular direction, it does not cost energy to change the orientation of the spontaneous curvature (similar to usual buckling). (c) “Phase diagram” in the plane $v_0$ vs. $\sigma$. In the elastic state, $B_{\text{eff}} > 0$ and the energy has its minimum at $\kappa = 0$. In the toro-elastic state, $B_{\text{eff}} < 0$ and an energy minimum at finite $\kappa$ emerges. In the coexistence region $\kappa = 0$ re-emerges as a local minimum, and the toro-elastic state is separated from it by a barrier. Parameters: $\sigma$ in nm and $v_0$ in $\frac{k_B T}{\text{nm}}$ for fixed $l_0 = 25$ nm. Virus radius $R = 50$ nm; bending rigidity $B = 3 k_B T \mu\text{m}$.</figcaption></figure>

Apart from its magnitude and short ranged nature, the detailed form of $v$ turns out to be non-essential for the phenomenology and emergence of the toro-elastic state. An instructive potential is the linear-exponential one,

$$v(l) = -v_0 \frac{l}{\sigma} \exp \left( 1 - \frac{l}{\sigma} \right), \quad (3)$$

as sketched in Fig. 3a. It has a minimum, corresponding to the bound state, at $l = \sigma$ with $v(\sigma) = -v_0$ and quickly flattens for $l > \sigma$. In the SI we also discuss a Gaussian potential, which has the width of the well as an additional adjustable parameter. While the latter makes the scenario more complex, it still exhibits the same states as now discussed for the potential given by Eq. (3).

Inserting Eqs. (1) and (3) into Eq. (2), the energy can be calculated analytically [19]. Importantly, depending on parameters, it can have the shape of a Mexican hat, see Fig. 3b: this is the “toro-elastic” state, where a finite curvature is preferred whose direction (i.e angle $\phi$) is ar-
bitrary and hence corresponds to a soft mode. Fig. 3c shows the different energy landscapes/shapes when varying the depth, $v_0$, and the minimum position, $\sigma$, of the potential for fixed spike distance $l_0$ and virus stiffness $B$. This phase diagram displays: (i) an elastic state, where the stable state is a single point $\vec{\kappa} = 0$ corresponding to a straight virus; (ii) the toro-elastic region, where the stable state space is a circle with $|\vec{\kappa}| = \kappa_{tor}$; and (iii) a coexistence region, where the (locally) stable state space is a circle and the origin, with a barrier to reach the toro-elastic state.

To understand when the straight state, $\kappa = 0$, becomes unstable we expand the energy up to second order in curvature, yielding $E = E_0 + \frac{1}{2}B_{eff}\kappa^2$ with $E_0 = -\frac{2\pi l_0 v_0}{\sigma}e^{1-\frac{l_0}{\sigma}}$ and an effective bending stiffness $B_{eff} = B + B_{bind}$ with

$$ \frac{B_{bind}}{2\pi} = v_0 l_0^2 e^{1-l_0/\sigma} \left[ \frac{R^2}{\sigma^2} \left( 1 - \frac{l_0}{2\sigma} \right) + \frac{l_0}{12\sigma} \left( 1 - \frac{l_0}{\sigma} \right) \right]. $$

This binding-induced part of the effective stiffness can become negative. In fact, this condition can be always satisfied for $l_0 \gtrsim 2\sigma$ if in addition the interaction strength is sufficiently large, $v_0 \gg e^{l_0/\sigma}B/l_0^2$: then $B_{bind}$ overrules $B$ and the filament curves with higher order terms stabilizing a finite toroidal curvature $\kappa_{tor}$. This toro-elastic state generically appears for strong surface interaction and sufficiently large (but not too large) spike spacing vs. interaction distance, cf. also the study of the more general Gaussian potential in the SI.

The curvature of the toro-elastic state adjusts such that a significant portion of the cross-section (typically half of it), benefits from the surface interactions. For that, the elements of the surface need to shift from their unperturbed distance $l_0$ to $\sigma$, i.e. by an amount $l_0 - \sigma \sim \kappa_{tor}Rl_0$, which yields the following estimate for the curvature of the toro-elastic state

$$ \kappa_{tor} \sim R^{-1} (1 - \sigma/l_0). \qquad (4) $$

The stiffness of the toroelastic state (i.e. the curvature in the rim of the Mexican hat, cf. Fig. 3b, along the radial direction), is dominated by the surface interactions. Assuming a curvature variation $\delta\kappa$ around $\kappa_{tor}$, such that the binding distance changes by $\sim \sigma$, we have $\delta\kappa \sim \frac{\sigma}{Rl_0}$ and the binding energy varies by $\delta E \approx |E_0|$. Equating $\frac{1}{2}K_{tor}(\delta\kappa)^2 \approx |E_0|$ then allows to estimate the stiffness of the toroelastic state to be

$$ K_{tor} \approx 4\pi \left( \frac{l_0}{\sigma} \right)^3 e^{1-\frac{l_0}{\sigma}} R^2 v_0 \qquad (5) $$

up to subdominant contributions from the bending elasticity.

Considering numbers, typical length scales are $l_0 = 25$ nm, $\sigma = 10$ nm and virus radius $R = 50$ nm; a reasonable estimate for the contact potential strength is

$v_0 = 1 \frac{k_B T}{\text{nm}}$. The bending rigidity of a virus is hard to measure [20], but considering the virus as a membrane tube we estimate $B = 2\pi R \cdot 10k_B T \approx 3 k_B T \mu\text{m}$. Using these values, one finds $B_{bind} \approx -5.7 k_B T \mu\text{m}$, hence indeed $B_{eff} < 0$ is at reach. Further we get $\kappa_{tor} \approx (100\text{nm})^{-1}$ estimated from Eq. (4); the toro-elastic stiffness, $K_{tor} \approx 100 k_B T \mu\text{m}$, is much larger than the usual bending rigidity.

**Toroelastic state driven by ligand dynamics.** Having seen how the toro-elastic state arises, we now show how it can be driven in an out-of-equilibrium situation. We assume that the virus is embedded in a solvent that contains a substantial concentration of glycan coated polymers (as in the mucus), which we model here for simplicity as di-glycan "dumbbells". If these glycans attach to two neighboring spikes, they introduce subtle force imbalances on top of the toro-elastic state. Moreover, these imbalances are dynamic due to the enzymatic cutting ability of the HEF spikes.

We assume the di-glycans to be harmonic polymer springs with spring constant $S$ and with a vanishing preferred length. They can be unbound (state $B_0$), bound to one spike (state $B_1$) or to two neighboring spikes (along the backbone), defining the double bound state $D$, see the upper part of the reaction scheme in Fig. 4a. In this double bound state, the glycans stretch out to a non-zero actual length as already given by Eq. (1), with $\phi$ the orientation of the curvature vector $\vec{\kappa}$ and $\theta$ the angular position of the spike it attaches to in the considered cross-section. In general, a cross-section will have an azimuthal imbalance of the (double-) bound density $D(\theta)$. The total glycan stretching energy per cross-section for a virus segment of length $l_0$ then reads

$$ E_{str}(D) = \frac{S}{2} \int D(\theta) l^2(\kappa, \theta). \qquad (6) $$

Assuming small curvatures $\kappa R$, $\kappa l_0 \ll 1$ and applying a mode ansatz $D(\theta) = D_0 + D_s \sin\theta + D_c \cos\theta$, to linear order $E_{str}(D) \approx \frac{S}{2}l_0^2(D_0 - R\kappa(D_c \cos\phi + D_s \sin\phi))$. The corresponding torque (per $l_0$ section) is then given by $M_{dr} = -\frac{\partial E_{str}}{\partial \phi}$ and in the frame co-moving with the toro-elastic curvature evaluates to

$$ M_{dr} = \frac{S l_0^2}{2} R \kappa D_s = m_{dr} D_s. \qquad (7) $$

This is the driving torque from the inhomogeneously distributed double bound glycans, with $m_{dr}$ the characteristic torque scale. With $S \sim 0.01 - 0.1 \frac{k_B T}{\text{nm}^2}$, the above parameters and $\kappa = \kappa_{tor}$, $m_{dr}$ can be estimated to be $2 - 20 k_B T$, which is appreciable [21].

Beyond neighboring spikes getting linked by freely floating di-glycans, we now also consider the enzymatic cutting activity of the HEF spike. The initial equilibrium reaction scheme has to be extended by two more states, as sketched in the bottom part of Fig. 4a. First, a double bound glycan can be cut into two, to a state
<figure><img src="image_6.png" /><figcaption>FIG. 4. (a) (left) Interaction of tension-inducing polymeric ligands (dumbbell springs) with the virus spikes. (right) The considered reaction kinetics of equilibrium binding and catalytic cutting. (b) The toroidal mode's angular frequency (in units of s⁻¹) as a function of the cutting rate $k_{cut}$ (units s⁻¹). The red curve shows the analytical result, Eq. (10), obtained without friction, and the black curve shows the numerical result with realistic frictional torque coefficient $\xi_h/m_{drive} = 0.01$ (see SI for details). Parameters: glycan concentration $G = 3$ mM; $k_1 = 0.1$ mM⁻¹s⁻¹, $k_{-1} = 0.1$ s⁻¹ (for IV-A [22, 23]); $k_2^0 = 1$ mM⁻¹s⁻¹, $k_{-2} = 0.1$ s⁻¹ (estimated). (c) The uniform, shape invariant rotation of a short toroidal virus or an elongated helical virus with frequency $\omega$ leads to a swimming velocity $v$ with the kinematics of fluid vortex lines [25]-[27].</figcaption></figure>

$C_2$ that still blocks both spikes for further attachments. If one of these cut parts detaches, or if a single bound glycan is cut, this yields state $C_1$. Only if all cut parts have left, one recovers state $B_0$ and the spike can be “used again” for attachment. The full reaction scheme of Fig. 4a translates into

$$ 
\begin{aligned}
\frac{d}{dt}B_1 &= k_1 G B_0 - (k_{-1} + k_2 (\theta - \phi) + k_{cut}) B_1 + k_{-2} D, \\
\frac{d}{dt}D &= k_2 (\theta - \phi) B_1 - k_{-2} D - k_{cut} D, \\
\frac{d}{dt}C_1 &= k_{cut} B_1 + k_{-1} C_2 - k_{-1} C_1, \\
\frac{d}{dt}C_2 &= k_{cut} D - k_{-1} C_2,
\end{aligned}
\quad (8) $$

with an additional equation for $B_0$. This latter can be eliminated via $B_0 + B_1 + C_1 + C_2 + D = 1$, since all quantities are site population fractions. Note that all considered states are functions of $\theta$. Assuming a curved virus that is rotating with constant angular frequency $\omega$,

i.e. $\phi(t) = \omega t$, and $\kappa = \text{const.}$ as suggested by the high stiffness of the toroidal state estimated above, we can move into the co-moving frame, transforming $\phi \to 0$ and $\frac{d}{dt} \to \omega \frac{\partial}{\partial \theta}$.

Importantly, the double binding "on" constant $k_2(\theta)$ depends on the curvature – since di-glycans bind more easily at the inside of bent regions as they have to stretch more on the outside of the bend. For small curvatures $\kappa R$, $\kappa l_0 \ll 1$, one can relate the reaction rates on the two opposing sides (see SI)

$$ \frac{k_2(\theta)}{k_2(\theta + \pi)} = e^{\frac{S}{2} \frac{-l^2(\theta) + l^2(\theta + \pi)}{k_B T}} \approx 1 + \left( \frac{2S l_0^2}{k_B T} R \kappa \right) \cos \theta $$

yielding

$$ k_2(\theta) = k_2^0 (1 + \alpha \cos \theta), \quad \alpha = \frac{S l_0^2}{k_B T} R \kappa \quad (9) $$

where $\alpha$ is a dimensionless coupling constant, in the following assumed to be small for simplicity.

The equations for the steady state dynamics in the co-moving coordinate system can be solved by applying a mode ansatz, where as above for the driving torque, all the chemical species can have a constant, a cos- and a sin-mode, $X = X^0 + X^c \cos \theta + X^s \sin \theta$ for $X = B_1, D, C_1, C_2$. In the case of negligible fluid friction, and for fast binding of the second bond of the dumbbell, $k_2^0 \gg k_{others}$, the solution can be given analytically, see SI. One finds that $\omega = 0$ is always a solution, while solutions with finite angular velocity, given by

$$ \omega^2 = \frac{k_1 G k_{cut}}{2} \left( 1 + \sqrt{1 + \frac{8k_{-1}^2}{k_1 G k_{cut}}} \right) - k_{-1}^2, \quad (10) $$

bifurcate from this stationary branch for sufficiently rapid glycan cutting rate

$$ k_{cut} > k_{cut}^{c,0} = \frac{k_{-1}^2}{3k_1 G}. \quad (11) $$

It is also possible to solve the problem in presence of friction. Assuming a hydrodynamic (Stokes) rotational friction torque $M_h = \xi_h \omega = 4\pi\eta R^2 l_0 \omega$ and equating it with the driving torque, Eq. (7), one gets $k_{cut} = k_{cut}^{c,0} + k_{cut}^1(\eta)$ with a correction that is to leading order linear in fluid viscosity $\eta$, proportional to $\frac{R}{\alpha^2 S l_0 \kappa}$ and that in addition depends on the chemical rates, see SI for details. Fig. 4b) shows the angular velocity $\omega$ as a function of the cutting rate $k_{cut}$: the red curve shows (the positive branch of) Eq. (10) and the black curve are numerical results including the frictional torque. Since IV-C has not been characterized yet concerning its kinetic parameters, we had to use estimates for IV-A [22, 23]. Note that cutting rates are estimated very conservatively, as IV-A's neuraminidase can reach cutting rates of up to $k_{cut} = 15$ s⁻¹ [24].
**Helix formation.** If the virus spike lattice consisted of discrete uncoupled sections, the toro-elastic energy would only depend on the magnitude of the local curvature. Consequently, each cross-section would behave independently with its curvature vector pointing in random directions. However, in general one expects multiple interdigitated and mutually staggered sub-lattices of spike-pairs, as sketched in Fig. 2a as pairs 1-2 and 3-4, with each of the lattices experiencing a mean effective curvature over their length scale ($\sim l$). Assuming for simplicity two sub-lattices, I and II, the effective curvature stemming from sub-lattice I and felt by sub-lattice II is $\kappa_{II} = \left| \frac{1}{l} \int_{-l/2}^{+l/2} \vec{\kappa}_I(s) ds \right|$. For a curvature vector of constant modulus that slowly winds with a torsion rate $\tau \ll l^{-1}$ like $\vec{\kappa}'(s) = \kappa(\cos(\tau s)e_x + \sin(\tau s)e_y)$, the total energy of the two sub-lattices can be written as, cf. SI,

$$ \frac{E_{tot}(\kappa, \tau)}{K_{tor}} = \frac{1}{4}(\kappa_{tor} - \kappa)^2 + \frac{1}{4}\left[ (\kappa_{tor} - \kappa) + \kappa \frac{l^2 \tau^2}{24} \right]^2 . \quad (12) $$

The energy is positive definite, with a curved, untwisted ground state $\kappa = \kappa_{tor}$ and $\tau = 0$. The first term is similar to a classical bending energy, followed by an unusual torsion-bend coupling that gives rise to a quartic torsional energy for $\kappa = \kappa_{tor}$. Notably, for curvatures larger than optimal, $\kappa > \kappa_{tor}$, the torsion becomes bistable with two preferred states $\tau_{1,2} = \pm \sqrt{\frac{24}{l}} \sqrt{1 - \kappa_{tor}/\kappa}$. That is, we expect the untwisted toroidal ground state to turn into a bistable helix (of any handedness) upon additional external bending torques. This includes in particular the torques induced by the above-discussed glycan binding. In practice, the symmetry of the two helical states is likely to be broken by a geometrical chirality of the natural spike lattice – an effect neglected here for simplicity.

**Self-propulsion.** For a toroidal or helical object uniformly rotating its body along the toroidal mode, while keeping its shape invariant (see Fig.4c), we expect the swimming kinematics to be that of a line of fixed vorticity $\propto \omega$, that is self-advecting through the fluid [25]. In particular the swimming kinematics becomes in leading order independent of the (small) helical torsion and occurs at typical propulsive velocities of spinning tori $v \sim \omega\kappa R^2$ [26, 27]. Note that this shape invariant swimming contrasts usual body-reshaping swimmers (propagating bending waves), where the swimming speed depends on the undulation amplitude and wavelength but not on the radius [29].

In conclusion, we have seen that viruses possessing the toro-elastic state can attain a spontaneous rotation upon binding and cutting interactions with the glycans floating in the mucus. The predicted angular velocities associated with this dissipative rotary shape-flexing dynamics are comparable to the rolling angular velocities of surface attached viruses (fractions of $s^{-1}$) [12, 13], where the glycan cutting and binding rates both contribute to

the speed. The typical active swimming velocity of tens of nm/s is rather modest compared to swimming speeds of organisms – bacterial flagellae turn at 50 Hz in water, flagella-mediated body-reshaping in spirochetes has a similar frequency [29] – but is still of the order of one virus size per second. Hence it is preferable to being effectively immobilized in the extremely viscous, sticky environment. The proposed mechanism hence adds another possible motility mode to the virus’ repertoire, beyond ratchet-like motion [30] and rolling [11, 12].

**Acknowledgements.** IMK thanks Jens-Uwe Sommer for continuous discussion and support.

<footnote>* f.ziebert@thphys.uni-heidelberg.de</footnote>

<footnote>† kulic@unistra.fr</footnote>

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$$E = \frac{1}{2} B \kappa^2 - 2\pi v_0 \frac{a}{\kappa R} e^{(1 - \frac{a}{\kappa R})} (I_0(a) - \kappa R I_1(a)),$$

where $a = \frac{2R}{\sigma} \sin(\kappa l_0/2)$ and $I_0, I_1$ are the modified Bessel functions of the first kind.

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[21] Note that this value is large, considering that it acts on a short section of size $l_0$. As a reference, the rotational friction at an angular frequency $\omega = 2\pi s^{-1}$ in a rather viscous medium of $\eta = 0.1$ Pa s (100 times of water) for $R = 50$ nm is approximately $M \approx 0.05k_BT$, i.e. largely subdominant compared to the linker-induced torque $M_{dr}$ for not too low $D_s$.

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