# The Physics of Self-Rolling Viruses

Pedro A. Soria Ruiz1, Falko Ziebert1,2, and Igor M. Kulić3,4 1Institute for
Theoretical Physics, Heidelberg University, D-69120 Heidelberg, Germany
2BioQuant, Heidelberg University, D-69120 Heidelberg, Germany
3Institut Charles Sadron UPR22-CNRS, F-67034 Strasbourg, France
4Institute Theory of Polymers, Leibniz-Institute of Polymer Research, D-01069
Dresden, Germany

###### Abstract

Viruses are right at the interface of inanimate matter and life. However,
recent experiments [T. Sakai, et al., J. Virol. 92, e01522-17 (2018)] have
shown that some influenza strains can actively roll on glycan-covered
surfaces. In a previous letter [F. Ziebert and I. M. Kulić, Phys. Rev. Lett.
126, 218101 (2021)] we suggested this to be a form of viral surface
metabolism: a collection of spike proteins that attach to and cut the glycans
act as a self-organized mechano-chemical motor. Here we study in more depth
the physics of the emergent self-rolling states. We give scaling arguments how
the motion arises, substantiated by a detailed analytical theory that yields
the full torque-angular velocity relation of the self-organized motor.
Stochastic Gillespie simulations are used to validate the theory and to
quantify stochastic effects like virus detachment and reversals of its
direction. Finally, we also cross-check several approximations made previously
and show that the proposed mechanism is very robust. All these results point
together to the statistical inevitability of viral rolling in presence of
enzymatic activity.

##  I Introduction

One of humanity’s greatest inventions is the wheel and technological
revolutions were carried by its ”motorization”. Being technologically so
indispensable, we might ask for the wheel’s utility in biology [1]. On the
macroscale examples are scarce, yet in the micro realm passive rolling, e.g.
of white blood cells [2] or malaria-infected red blood cells [3], occur in
shear flow and are important for the functioning of the immune response and
the traveling of the parasite through our body, respectively. Active self-
propelled rolling was unknown for a long time. But surprisingly, the motorized
wheel was rolling also in nature for ages: our old molecular adversary – the
influenza virus – apparently is able to use its whole body as a chemically
driven monowheel that actively rolls on our lung cells’ surfaces by
catalytically hydrolyzing sugars sticking out from the cell membranes. This
surprising (and maybe even alarming) phenomenon of active virus surface-
rolling has been demonstrated first by Sakai et al. [4, 5] and interpreted as
a Brownian-ratchet-like effect. It has been also observed indirectly [6] and
is discussed now as an important pathway helping the virus to cross and
navigate the mucus [7, 8].

The underlying physical mechanism – different from the classical Brownian
burnt bridge model [9, 10] – has been proposed recently in [11], where we
outlined elements of a model which we elaborate deeper in the present work.
The initial model appears to have left parts of the molecular motor community
in slight disbelief [12] whether the mechanism could actually work as
described. Here we explain the robustness and inevitability of the rolling
state as proposed earlier and verify approximations made in [11] against more
detailed analytical calculations and stochastic simulations. In addition we
develop a scaling view on the mechanism. We study stochastic effects such as
reversals of direction and virus detachment. And finally, we clarify how the
directional stability and processivity physically emerges by pinpointing an
internal “mechano-chemical flywheel” – a long-living internal polarization
mode with directional memory – that allows for highly persistent rolling to
occur in spite of large external noise and at zero Reynolds number.

The paper is organized as follows: in section II we describe the basic
ingredients, from reaction kinetics to force balance. Section III then goes on
to explain on an intuitive level why the virus actually rolls and how its
rolling steady state arises. In section IV the steady rolling is studied via
simple approximations allowing the force-velocity relation to be analytically
investigated. Section V then describes a stochastic version of the model which
we show to be consistent with the continuum version in the steady state.
Beyond this limit, we also explore stochastic phenomena, like the rates of
virus detachment, reversals, run lengths etc. Section VI critically
scrutinizes approximations made so far and quantifies the dynamical
persistence of the mechanism by studying a virus being stopped
instantaneously, leading to a build-up of torque via the “flywheel” effect.
Finally, in section VII we discuss implications of the model for
biology/virology and we conclude with some open questions and experimental
tests.

##  II Basic Model

Figure 1:  Influenza A has two spike proteins on its surface: hemaglutinin
(HA, blue) which attaches to sugar residues from short surface-bound glycan
chains (green) and neuraminidase (NA, red) which cuts the sugar residues such
that they are no longer available for HA binding. In a first step, we consider
a virus rolling under an externally applied force over a glycan-covered (cf.
green dots) surface, as sketched in A) as both a cross-section view and a top
view. The HA-glycan binding kinetics (in the contact zone as sketched in the
left panel by the bar) will result in a friction force opposing the rolling
motion. B) In a second step, we consider the effect of NA cutting. The dynamic
self-organization (self-polarization) of the bound linker profile in the
contact zone now allows for steady self-rolling. The NA “consumes” glycans
which leads to a depleted trail on the back of the virus (see right panel).

We first present a mean-field description that will give much general insight
into the problem and, using several approximations, allows for a detailed
analytical treatment of the steady rolling.

The results derived here are formulated for any viral capsid cross-section
orthogonal to the rolling direction and are equally valid for both influenza
isoforms [13, 14]: the cylindrical (filamentous) and the spherical virus
geometry. The main difference between the two cases is the system size (total
number of interacting spike proteins) and the enhanced role of fluctuations
for spherical or smaller aspect ratio viruses. Stochastic effects are
postponed to section V, where we also critically scrutinize the validity of
the assumptions made here.

###  II.1 Influenza spike proteins and their kinetics

As one of the omnipresent molecular adversaries of mankind, the influenza
virus (IV) and its proteins have been extensively characterized [15, 16, 17,
18]. As often in the virus realm, influenza is in fact a whole family of
viruses that have evolved slightly differently. We will focus here on the two
viruses where motility has been evidenced experimentally [4, 5], namely
influenza-A (IVA) and influenza-C (IVC).

IVA has two spike proteins that interact with the host membrane: Hemaglutinin
(HA) and Neuraminidase (NA), see Fig. 1. These are distinct $\sim 10\,$nm
sized entities (HA is a trimer, NA a tetramer) that perform two distinct and
mutually competing functions: HA binds to sialic acid residues of glyco-
peptides and lipids coating the surface of our cells, while NA acts
antagonistically by hydrolytically cutting the same sialic acid residue that
HA binds to. Importantly, for steric reasons the residue can be either bound
by one HA or by one NA molecule, but not by both at the same moment. In IVC,
the two proteins are fused together into a single spike protein [19], meaning
the “attaching spike” and the “cutting spike” are co-localized. But again,
only one of the binding sites can interact with a glycan residue at a time.
Note that this renders IVA more flexible, as it can for instance polarize its
HA-NA distribution on its surface and engage in other mechanisms of motion
than described here, as discussed previously [20]. This motion however is a
much slower process, cf. section VII. Due to lack of available experimental
data, we study here the kinetics with IVA parameters and assume that the
values for IVC (once available) should be of similar order.

The binding/unbinding kinetics of HA will be described via on/off rates
$k_{on}$, $k_{off}$. These have been characterized experimentally [21, 22]
yielding a dissociation constant $K_{d}=\frac{k_{off}}{k_{on}}=1$-$5\,$(we use
2) mM, $k_{off}=10^{-1}$-$1\,(1)\,{\rm s}^{-1}$ and
$k_{on}=0.01$-$1\,(0.5)\,{\rm mM}^{-1}{\rm s}^{-1}$. NA transiently binds
(with rates $k_{1}$, $k_{-1}$) and then enzymatically cuts the sialic residues
with a rate $k_{cut}$, making the glycans irreversibly inactive for HA
binding. We hence use a Michaelis Menten description with a Michaelis constant
$K_{M}=(k_{-1}+k_{cut})/k_{1}$. NA’s enzymatic activity has also been measured
[23] to yield $K_{M}\simeq 14.3$ mM and $k_{cut}\simeq 15\,{\rm s}^{-1}$,
implying $V_{cut}=k_{cut}N_{NA}=15\,{\rm mM}{\rm s}^{-1}$ for a typical NA
concentration of $N_{NA}=1$ mM.

###  II.2 Contact interval

During virus rolling all the force generation happens in the virus-substrate
contact region, i.e. the interface where the virus and the glycan-coated
substrate (cell membrane) meet. Here we roughly estimate the size of this
region for a cross-section of a cylindrical (or spherical) virus and will
later substantiate the result by considering the full binding kinetics, cf.
section VI.1.

We assume that glycan chains are present at a high concentration $G_{0}$ –
well in excess to spike proteins (throughout this work we use $G_{0}=10\,{\rm
mM}$ and for the HA spike concentration $H_{0}=2\,{\rm mM}$, as estimated
earlier [11]). The glycan chains are constantly binding to and unbinding from
the HA spike proteins and elongate to a length $l=R(1-\cos\phi)$ in that
process, where $\phi$ is the angle measured from the virus symmetry axis and
$R$ the virus radius. If bound they gain a free energy

$\Delta G=k_{B}T\ln\left(\frac{G_{0}}{K_{d}}\right)$ (1)

with $K_{d}$ the dissociation constant. In turn they have to pay the elastic
energy of getting stretched

$E_{el}\left(\phi\right)=\frac{S}{2}l^{2}\simeq\frac{SR^{2}}{8}\phi^{4}\,\,\,{\rm
for}\,\,\,\phi\ll 1,$ (2)

where glycan chains were considered ideal linear springs with spring constant
$S\sim 0.01$-$1\,k_{B}T/{\rm nm}^{2}$ \- a typical range for polymers of few
nm length (we chose $0.1k_{B}T/nm^{2}$). Note that the cylindrical (or
spherical) geometry of the virus results in a strong dependence on the angle
$\phi$ and on the virus radius $R$ (which for IV is typically $\simeq 50$ nm).
We neglect here the effect of NA binding for simplicity, since it is short
lived compared to HA.

Balancing the two energy terms yields the typical angular size of the contact
zone with $\phi\in[-\phi_{c},\phi_{c}]$ to be

$\phi_{c}=\left(\ln\left(\frac{G_{0}}{K_{d}}\right)\frac{8kT}{SR^{2}}\right)^{1/4}.$
(3)

In the following we assume the contact area size to be a constant, even when
the virus is rolling.

###  II.3 Torque balance

The torque of an attached virus can be calculated using the stretching force
per linker $F_{el}=-\frac{\partial}{\partial l}E_{el}$ yielding a torque
$\propto SR^{2}\left(1-\cos\phi\right)\sin\phi\simeq\frac{1}{2}SR^{2}\phi^{3}$
for small angles. If the linkers have an angular density
$\rho_{HA}\,b\left(\phi\right)$ with $\rho_{HA}$ the angular density of HA
spikes and $b\left(\phi\right)$ the angular probability density of each linker
being bound, the total torque acting on the virus is just the integral over
all bound linkers,

$m=-m_{0}\int_{-\phi_{c}}^{+\phi_{c}}b\left(\phi\right)\phi^{3}d\phi,$ (4)

with $m_{0}=\frac{1}{2}SR^{2}\rho_{HA}$ the characteristic torque scale.

When the virus is rolling at typical angular speeds, experimentally
$\omega\simeq 1\,{\rm s}^{-1}$ [5], using typical densities of linkers one can
estimate all other torques, e.g. from hydrodynamics, to be negligible.
Therefore the torque balance $m=0$ has to hold (to very good approximation) at
all times.

Note that for simplicity we assumed here that there is no compression of the
chains by the virus – except at $\phi=0$ to fulfill force-balance. This
simplification can be relaxed as explained in appendix B.

###  II.4 Dynamics of bound linkers and free glycans

The binding of the HA spikes to the glycans can be described by a simple on-
off kinetics. Denoting the bound HA-glycan linker concentration by $B$, the
unbound (open) HA concentration by $O$ and the free glycans by $G$, one has
$\partial_{t}B=k_{on}G\,O-k_{off}B$ and an equation with opposite signs on the
r.h.s. for $\partial_{t}O$. Obviously, as $O+B=H_{0}$ with $H_{0}$ the total
number of HA, one can immediately eliminate the equation for the open HA.

Adding the dynamics for the free glycans, one can write

$\displaystyle\partial_{t}B+\omega\partial_{\phi}B$ $\displaystyle=$
$\displaystyle\,\,\,\,\,k_{on}G\left(H_{0}-B\right)-k_{off}B\,,$ (5)
$\displaystyle\partial_{t}G+\omega\partial_{\phi}G$ $\displaystyle=$
$\displaystyle-k_{on}G\left(H_{0}-B\right)+k_{off}B-f_{cut}\,,\,\,\quad$ (6)

were we accounted for a (potential) rolling with angular velocity $\omega$,
leading to advection of all concentration profiles as reflected by the second
term on the l.h.s., and for the enzymatic cutting of $G$ by the NA spikes as
reflected by the total cutting rate $f_{cut}$ acting as a sink.

It is important to note that we made the approximation that the on/off-
kinetics of HA-glycan binding satisfies $\frac{k_{off}}{k_{on}}=K_{d}$. That
is, we neglected that $K_{d}$ is in general stretching force- [24] and hence
angle-dependent. This – rather violent-looking – approximation allows for an
analytical treatment. After having understood the general mechanism of virus
rolling, we show in section VI.1 that the angle-dependence can be included –
in both continuum numerics and stochastic simulations – and that this proper
account of the detailed balance does not change the behavior qualitatively. In
other words, at this stage, in the simple model we use the stretch-dependence
only to determine the size of the contact interval, but not for the kinetics.
Hence the glycan-binding profile of a static virus will be box-like (constant
in the contact interval and zero outside) while in reality it decays with
$\exp(-\phi^{4})$, cf. section VI.1.

The total cutting rate $f_{cut}$ can be approximated by a Michaelis-Menten
kinetics, as discussed in section II.1, with a Michaelis constant $K_{M}$ and
a cutting velocity of the sialic acid (glycan) residues,
$V_{cut}=k_{cut}N_{NA}$ set by the enzymatic turnover rate $k_{cut}$ and the
enzyme concentration $N_{NA}$, giving
$f_{cut}=\frac{V_{cut}G}{K_{\mathrm{M}}+G}$.

Finally, the minimal model reads

$\displaystyle\partial_{t}B+\omega\partial_{\phi}B$ $\displaystyle=$
$\displaystyle k_{on}G\left(H_{0}-B\right)-k_{off}B,\,$ (7)
$\displaystyle\partial_{t}G+\omega\partial_{\phi}G$ $\displaystyle=$
$\displaystyle-
k_{on}G\left(H_{0}-B\right)+k_{off}B-\frac{V_{cut}G}{K_{\mathrm{M}}+G}\,\,\,\quad$
(8)

and has to be solved on the contact interval $\phi\in[-\phi_{c},\phi_{c}]$
with $\phi_{c}$ given by Eq. (3) and together with the torque balance
constraint, i.e. $m$ defined in Eq. (4) must be zero at all times.

##  III Steady rolling - scaling and numerics

Let us now discuss the just proposed – deterministic and mean field-type –
model in the steady state on the scaling level. In steady state, the virus is
either not moving at all or rolls with constant angular velocity $\omega$. The
first main question is: are states with $\omega\neq 0$ possible and if so to
understand and analyze the causing mechanism.

Figure 2: Sketches of rolling virus cross-sections with representations of
their bound linker distributions within the contact zone. Panels (1), (2):
passive case (no NA activity) for low vs. high angular velocity $\omega$. Here
the rolling must be due to an externally applied force and the linker-induced
torque, $m$, counteracts the rotation. Panels (3), (4): active case with
enzymatic NA activity, again for low and high $\omega$. The linker-induced
torque accelerates (3) or decelerates (4) the rotation. Panel ($3_{1/2}$):
steady state, where the two regions at the front and back counteract and the
total torque is zero. (5) Zoom of the passive (blue dashed) and active (red)
profiles with characteristic profile slopes $\alpha$ and $\beta$ (see text)

###  III.1 Scaling arguments

Before diving into the detailed calculations, let us try to explain the
rolling motion using simple scaling arguments. Our focus lies on what the
bound linker distribution looks like in the contact zone and the effects
thereof for the torque balance.

First consider the simplest case of a virus that is forced to roll by an
externally applied torque – as sketched in Fig. 1A) – and that does not have
enzymatic NA activity, i.e. only binds its HA to the glycans on the surface.
In rolling direction, the virus encounters unbound glycans and the HA needs
time to bind, which means the bound HA-glycan distribution in the co-moving
virus frame has the shape as sketched in Fig.2(1): it increases from zero with
a certain slope and levels off at a plateau value $B_{pl}$, corresponding to
the mean bound linker distribution a non-moving virus would have. The slope is
determined by two quantities: first, the linker attachment rate

$\alpha=k_{on}H_{0}G_{0}$ (9)

which is linked to the plateau value by $\alpha t_{pl}=B_{pl}$, with $t_{pl}$
the time it takes to establish the equilibrium plateau. The second is the
actual rolling speed of the virus $\omega$. Since $\omega t\sim\phi$ the slope
in the angular distribution $B(\phi)$ is in fact given by
$\frac{\alpha}{\omega}$. Consequently, one expects the slope to be steep for a
slowly rolling virus and shallow if the virus rolls very fast, cf. Fig. 2(1)
vs. (2).

Now let us add a weak enzymatic cutting by NA. In rolling direction, the
binding kinetics will still dominate, for weak NA cutting rate $\omega$. Hence
the binding-induced positive slope in the attachment-region prevails. For a
steady rolling virus, however, with the distance from the front, the NA has
linearly more time to cut off the glycans and hence one expects that the
second, plateau region is transformed into an approximately linear, negative
slope which we denote by $\beta$. This slope has in general a complicated
parameter dependence, but it will be roughly proportional to the enzymatic
cutting velocity

$\beta\propto V_{cut}.$ (10)

By the same argument as before, the slope in the angular distribution
$B(\phi)$ is given by $\frac{\beta}{\omega}$ and the slope at the back is
steep for a slowly and shallower for a rapidly rolling virus, cf. Fig. 2(3)
vs. (4). Fig. 2(5) shows a zoom on the contact zone with the slopes labeled
and with sketches of their different origins.

What are the consequences of these linker distributions for the overall
torque? Note that a homogeneous, constant distribution $B(\phi)$ (and likewise
any symmetric one) is perfectly balanced. Hence the linkers always missing at
the front due to the time it needs to establish the equilibrium distribution
imply that there is an excess force acting on the back, resulting in a torque
counteracting the rotation, as indicated e.g. in Fig. 2(1) where the torque
$m$ is acting against $\omega$. Consequently, the attachment dynamics results
in an effective friction. In turn, the second, negative slope implies the
opposite: an excess force on the front, accelerating the motion. Note that
this is not forbidden thermodynamically: the action of NA is an active process
consuming, or rather cutting in an irreversible manner, the glycans. Now it
depends on the relative slopes/relative sizes of the two regions whether the
virus is overall accelerating, as in Fig. 2 (3), or decelerating as in Fig. 2
(4). In between there is the possibility of a stationary state where the
overal torque is zero, as sketched in Fig. 2($3_{1/2}$). This exactly
corresponds to the searched-for steady rolling state in the absence of
external driving.

We can substantiate this argument by estimating the torque-angular velocity
relation. This relation is a nonlinear function – and hence allows for non-
trivial steady states – due to the different $\omega$-dependencies of the two
regions, the attachment-dominated front and the cutting-dominated back. In
fact, for the attachment dynamics, if the respective region is small, the
torque evaluates to
$m_{att}\propto-\frac{\alpha}{\omega}\cdot\phi_{pl}^{2}\cdot\phi_{c}^{3}$,
where the first term is the profile’s slope, the second term arises from the
integral over the linear slope up to $\phi_{pl}$, the angle where the plateau
begins, and $\phi_{c}^{3}$ is related to the geometric lever arm. The minus
sign indicates the direction, opposed to the rotation. Since $\phi_{pl}=\omega
t_{pl}$ one gets $m_{att}\propto-\phi_{c}^{3}\,\alpha\,\omega$, i.e. a
“Stokesian hydrodynamics”-like friction linear in $\omega$. In contrast, for
the cutting dynamics the contribution of the small front region does not
matter as cutting takes place everywhere at a uniform rate. Hence the torque
integral over the region of cutting is
$m_{cut}\propto\frac{\beta}{\omega}\cdot\phi_{c}^{2}\cdot\phi_{c}^{3}$, i.e.
depends on $\omega$ only via the slope. The positive sign indicates its
accelerating effect discussed above. Clearly, the overall torque balance
$0=m=m_{att}+m_{cut}$, allows for steady state solutions of the type

$\omega\propto\pm\phi_{c}\sqrt{\frac{\beta}{\alpha}}\propto\sqrt{V_{cut}}.$
(11)

That is, the steady state rolling velocity grows rapidly for small cutting
velocities $V_{cut}$.

###  III.2 Time-angle correspondence and numerical solution

Let us confirm numerically that self-rolling is possible. In the steady state
we can drop the time derivative in Eqs. (7), (8) and have to solve

$\displaystyle\omega\partial_{\phi}B$ $\displaystyle=$ $\displaystyle
k_{on}G\,\left(H_{0}-B\right)-k_{off}B\,$ (12)
$\displaystyle\omega\partial_{\phi}G$ $\displaystyle=$ $\displaystyle-
k_{on}G\,\left(H_{0}-B\right)+k_{off}B-\frac{V_{cut}G}{K_{\mathrm{M}}+G}\,,\,\,\quad$
(13)

combined with $b(\phi)=B(\phi)/H_{0}$ giving the closure condition in terms of
the torque balance, Eq. (4)

$m=-\frac{m_{0}}{H_{0}}\int_{-\phi_{c}}^{+\phi_{c}}B\left(\phi\right)\phi^{3}d\phi=0\,.$
(14)

To do so, we can apply the following strategy, which could be called “time-
rotation angle correspondence”. The angular advection operator,
$\omega\partial_{\phi}$, and the time derivative operator, $\partial_{t}$, can
be treated on the same grounds by replacing time with the angle scaled by the
angular frequency: $t\rightarrow\frac{\phi+\phi_{c}}{\omega}$ (or $\phi=\omega
t-\phi_{c}$). The time window corresponding to passing the contact interval
$[-\phi_{c},\phi_{c}]$ is then $[0,T]$ with $T=\frac{2\phi_{c}}{\omega}$.

This observation suggests to think of the steady state as a dynamic relaxation
of the concentrations on the time interval $[0,T]$. One can hence treat the
problem as an initial value problem at one boundary of the interval (the one
in rolling direction) with $B\left(0\right)=0$ and $G\left(0\right)=G_{0}$ (in
the not yet visited region, nothing has bound and no glycan has been consumed
yet): the solution of the dynamic problem $B_{dyn}\left(t\right)$ on a time
interval $[0,T_{test}]$ can be obtained without any reference to the actual
angular frequency $\omega$.

In a second step $B_{dyn}\left(t\right)$ can be used to obtain the torque by
evaluating

$m=-\frac{m_{0}}{H_{0}}\int_{0}^{t_{up}=2\phi_{c}/\omega}B_{dyn}\left(t\right)\left(\omega
t-\phi_{c}\right)^{3}\omega dt$ (15)

Note that high frequencies correspond to taking the integral over a short time
interval and vice versa. Given the curve $B_{dyn}\left(t\right)$ on a large
enough interval, $[0,T_{test}]$ with $T_{test}>T$ for all considered $\omega$
values, one can now scan the upper boundary $t_{up}=2\phi_{c}/\omega$ by
varying the angular frequency and hence determine $\omega$ such that
$m\left(\omega\right)=0$, which is a simple root finding problem. Solving
along the same lines for $m\left(\omega\right)=m_{ext}$, with an external
torque $m_{ext}$, allows also to obtain the torque-angular velocity relation.

Figure 3:  a) Angular velocity $\omega$ of the steady rolling state as a
function of NA concentration $N_{NA}=[NA]$. One abtains a square-root
behavior, $\omega\propto\sqrt{N_{NA}}$, as suggested by scaling. b) Adding an
external torque $m_{ext}$ allows to obtain the torque-angular velocity
relation, which is a nonlinear function of $\omega$. Three different values of
NA concentration are shown, the respective free rolling velocities (for
$m_{ext}=0$) are also indicated in a).

Fig. 3 a) shows the angular velocity of the steady rolling state, obtained
numerically, as a function of enzymatic activity, i.e. NA concentration. One
clearly sees the square-root behavior as obtained by scaling,
$\omega\propto\sqrt{\beta}$ with $\beta\propto V_{cut}\propto N_{NA}$. Fig. 3
b) shows the torque-angular velocity relation obtained numerically for three
values of NA concentration. Free rolling corresponds to $m_{ext}=0$.

The results obtained numerically and shown in Fig. 3 are intriguing: in fact,
any – even a small – enzyme activity leads to finite rolling motion in this
mean field-type model. The response of the motor close to free rolling is as
expected – an assisting torque speeds up the rolling and a counter torque
slows it down, but there is a strong nonlinear dependence for higher counter
torques. Both these findings can be understood analytically and can be related
to the parameters of the spike dynamics, as shown in the next section. They
will then be critically compared to a model including stochastic fluctuations
in section V.

##  IV Steady rolling - Analytical theory

In the steady state much analytical insight can be gained using the main ideas
introduced in the last chapter, i.e. the time-angle-correspondance and the
”line-approximations” of the linker profiles, as sketched in Fig. 2.

###  IV.1 Analytical solution for forced rolling without enzymatic activity

We first consider the passive case, i.e. in absence of catalytic activity.
That is, we assume that the virus is forced to roll with a given steady-state
angular velocity $\omega$ by a weak, externally applied torque. For
$V_{cut}=0$ and in the steady state, Eqs. (12, 13) imply the conservation law
$\partial_{\phi}\left(B+G\right)=0$. Assuming a homogeneous initial glycan
coverage $G_{0}$, we can hence rewrite $G(\phi)=G_{0}-B(\phi)$. This reduces
the problem to a single equation

$\omega\partial_{\phi}B=k_{on}\left(G_{0}-B\right)\,\left(H_{0}-B\right)-k_{off}B$
(16)

which, for the initial condition $B(-\phi_{c})=0$ (rolling to the left), can
be solved exactly

$B\left(\phi\right)=\frac{C_{0}-C_{1}}{2}-\frac{C_{1}}{\frac{C_{0}+C_{1}}{C_{0}-C_{1}}e^{\frac{C_{1}k_{on}}{\omega}\left(\phi+\phi_{c}\right)}-1}.$
(17)

Here $C_{0}=H_{0}+G_{0}+K_{d}$, $C_{1}=\sqrt{C_{0}^{2}-4H_{0}G_{0}}$ are
constants determined by the concentrations and reaction kinetics. The solution
can also be given in time-domain:

$B\left(t\right)=\frac{C_{0}-C_{1}}{2}-\frac{C_{1}}{\frac{C_{0}+C_{1}}{C_{0}-C_{1}}e^{C_{1}k_{on}t}-1}.$
(18)

The resulting bound linker profile on the contact angle interval
$[-\phi_{c},\phi_{c}]$ is shown in Fig. 4, cf. also the cases sketched as (1)
and (2) in Fig. 2. It is characterized by an increase of the bound HA-glycan
links leveling at a plateau value of

$B_{pl}=\frac{C_{0}-C_{1}}{2}.$ (19)

For simplicity, and to be able to proceed with a perturbative approach taking
enzymatic activity into account, we approximate the exact profile by two
lines: first, in the region of its rapid increase, $B$ is approximated by the
slope at the front, and in the second region by its plateau value. This is
most transparent in time-space where one has

$B\left(t\right)=\begin{cases}\alpha t&\text{ for $0\leq t\leq t_{m}$}\\\
\alpha t_{m}&\text{ for $t_{m}\leq t\leq T$}\end{cases}$ (20)

for $t\in\left[0,T\right]$ with $T=\frac{2\phi_{c}}{\omega}$ as before and

$\alpha=k_{on}H_{0}G_{0}\,,\,\,\,t_{m}=\frac{B_{pl}}{\alpha}$ (21)

with $\alpha$ the initial slope (i.e. the linker binding velocity) and $t_{m}$
the time needed to reach the plateau/maximum (cf. $t_{pl}$ discussed in
section III.1). In angle-space one has

$B\left(\phi\right)=\begin{cases}\frac{\alpha}{\omega}(\phi+\phi_{c})&\text{
for $\phi\in[-\phi_{c},\phi_{pl}]$}\\\
\frac{\alpha}{\omega}(\phi_{pl}+\phi_{c})=B_{pl}&\text{ for
$\phi\in[\phi_{pl},\phi_{c}]$}\end{cases}$ (22)

with $\phi_{pl}$ the angle where the plateau is reached. This angle space view
is especially transparent to derive the scaling discussed earlier, cf.
appendix A.

Figure 4:  The blue curve shows the bound linker profile on the contact angle
interval $[-\phi_{c},\phi_{c}]$ as given by Eq. (17). The virus is forced to
roll to the left. The fact that the linkers need time to bind to the newly
encountered substrate is reflected by the increase on the left leveling to a
plateau, cf. Fig. 2 and the discussion is section III.1. The dashed lines show
the two-line approximation given by Eq. (22). Parameters as given in section
II, implying $\phi_{c}\simeq 0.5$.

###  IV.2 Multiple-line approximations

One can proceed the analysis by using the approximations of the bound linker
profiles by several lines. Typically, two lines are needed, cf. the cases
sketched in Fig. 2 as (1),(2),(4). When the linkers are completely cut at the
back, three lines are needed, cf. Fig. 2 (3). These multiple-line
approximations also allow us to treat the effect of enzymatic activity
analytically and to discuss the generic physics of the torque-angular velocity
relation associated with a given bound linker profile.

####  IV.2.1 Case of no enzymatic activity

This is the case just discussed where $B(t)$ is approximately given by Eq.
(20). The clear advantage of a line approximation is that we can calculate the
torque integral analytically. From Eq. (4) we have to evaluate

$m=-\frac{m_{0}\omega^{4}}{H_{0}}\int_{0}^{T}\left(t-\frac{T}{2}\right)^{3}B\left(t\right)dt$
(23)

with $T=\frac{2\phi_{c}}{\omega}$. Importantlly, we can write this as an
integral of the constant plateau value over the full range,
$\propto\int_{0}^{T}\left(t-\frac{T}{2}\right)^{3}B_{pl}\,dt=0$, which
vanishes due to symmetry, plus an integral over the first zone only (the one
at the front in rolling direction)

$m=-\frac{m_{0}\omega^{4}}{H_{0}}\int_{0}^{t_{m}}\left(t-\frac{T}{2}\right)^{3}\alpha(t-t_{m})dt\,.$
(24)

This highlights the fact that the torque arises from the imbalance/asymmetry
of the linker distribution associated with linkers having to form in rolling
direction in the region $0\leq t\leq t_{m}$.

Before solving in full generality, we can assume that the plateau is rapidly
reached, i.e. $t_{m}\ll T$, yielding

$\displaystyle m$ $\displaystyle\simeq$
$\displaystyle\frac{m_{0}\omega^{4}}{H_{0}}\alpha\left(\frac{T}{2}\right)^{3}\int_{0}^{t_{m}}(t-t_{m})dt=-\frac{m_{0}}{H_{0}}\frac{\phi_{c}^{3}}{2}\frac{B_{pl}^{2}}{\alpha}\omega,\quad\,\,\,$
(25)

where we used $T=\frac{2\phi_{c}}{\omega}$ and $t_{m}=B_{pl}/\alpha$. Eq. (24)
can also be integrated completely to yield

$m=\frac{m_{0}}{H_{0}}\frac{\alpha}{80}\omega^{4}T^{5}p\left(\frac{t_{m}}{T}\right)$
(26)

with the polynomial

$p\left(x\right)=-x^{2}\left(5-10\,x+10\,x^{2}-4\,x^{3}\right)$ (27)

for $t_{m}\leq T$; for $t_{m}>T$ one has to use $p(1)=-1$.

Let us discuss the limits. For very slow rolling, cf. Fig. 2(1), the first,
attachment-dominated regime shrinks with only the plateau left, and the torque
hence vanishes due to symmetry. For $t_{m}\ll T$, corresponding to
intermediate rolling speeds and sufficiently rapid build-up of the plateau,
one has $p(x)\simeq-5x^{2}$ which agrees with Eq. (25). Hence for intermediate
rolling speed one gets for the torque-velocity relation for passive rolling,

$m_{diss}\left(\omega\right)=-\xi_{diss}\omega\,\,,\,\,\,\,\xi_{diss}=\frac{m_{0}}{H_{0}}\frac{\phi_{c}^{3}}{2}\frac{B_{pl}^{2}}{\alpha}.$
(28)

This is a purely frictional torque, acting against the motion and linear in
$\omega$. The effective friction constant, $\xi_{diss}$, is determined by both
the slope, $\alpha$, and the plateau value, $B_{pl}$, of the bound linker
distribution, as well as the size of the contact interval $\phi_{c}$, which
themselves contain all system parameters.

For faster rolling, the friction is determined by $p(x)$. For very fast
rolling, faster even than in Fig. 2(2), the linking is so slow compared to the
rolling that there will be no plateau at all. In that case, one evaluates
$m=-\frac{m_{0}}{H_{0}}\alpha\omega^{4}T^{5}\frac{1}{80}$, which corresponds
to the value $p(1)$. The torque is hence still frictional and due to
$T=\frac{2\phi_{c}}{\omega}$ scales with $\omega^{-1}$. Hence in this regime,
the faster the rolling the less friction: less linkers bind because of limited
time and hence fewer resist the motion. Overall, friction first increases,
then is given by $p(x)$ and then decreases again.

####  IV.2.2 Case of weak enzymatic activity

In the presence of enzymatic NA activity, cf. Fig. 2(3)-(5), the first region
of increasing linker density is still present (in fact, only slightly
modified), while the plateau is transformed into a slowly decreasing function,
implying $B(t)$ having a maximum around $t_{m}$. Focusing on cases (4),(5)
where the glycan is not completely cut at the back, we can use the following
parameterization

$B\left(t\right)=\begin{cases}\alpha t&\text{ for $0\leq t\leq t_{m}$}\\\
\alpha t_{m}-\beta\left(t-t_{m}\right)&\text{ for $t_{m}\leq t\leq
T$}\end{cases}$ (29)

with two non-negative constants $\alpha>0,\beta\geq 0$ (with $\beta=0$
corresponding to the case without NA activity just discussed, without maximum
but a plateau). At the moment $\beta$ is just a parameter. In section IV.3 we
will determine it as a function of the underlying model parameters using
perturbation theory.

We can again bring the torque integral in a convenient form (using that the
integral over the plateau over the full contact area vanishes) and get

$\displaystyle m$ $\displaystyle=$
$\displaystyle-\frac{m_{0}\omega^{4}}{H_{0}}\left[\int_{0}^{t_{m}}\alpha(t-t_{m})\left(t-\frac{T}{2}\right)^{3}dt\right.$
(30)
$\displaystyle\left.\quad\qquad-\int_{t_{m}}^{T}\beta(t-t_{m})\left(t-\frac{T}{2}\right)^{3}dt\right]$

If the plateau is rapidly reached, $t_{m}\ll T$, one obtains

$\displaystyle
m=-\frac{m_{0}}{H_{0}}\left[\alpha\omega\frac{1}{2}\left(t_{m}\right)^{2}\phi_{c}^{3}-\frac{\beta}{\omega}\frac{2}{5}\phi_{c}^{5}\right].$
(31)

(this is also derived in Eq. (57)). Importantly, the second term has the
opposite sign – it is active – since now $t>t_{m}$ in the integration.
Secondly, the second term is independent of $t_{m}$ for small $t_{m}$, which
leads to the $1/\omega$ dependence.

Eq. (30) can again be integrated completely, yielding

$m=\frac{m_{0}\omega^{4}}{H_{0}}T^{5}\left\\{\frac{\alpha}{80}\,p\left(\frac{t_{m}}{T}\right)+\frac{\beta}{80}\left[p\left(\frac{t_{m}}{T}\right)+1\right]\right\\}.$
(32)

We hence get the relation

$\frac{80H_{0}}{\left(\alpha+\beta\right)\omega^{4}T^{5}}\frac{m}{m_{0}}=p\left(\frac{t_{m}}{T}\right)+\frac{\beta}{\alpha+\beta},$
(33)

from which one can draw general conclusions: First, $\omega=0$, implying
$T=\infty$ and hence $\frac{t_{m}}{T}=0$ and
$p\left(\frac{t_{m}}{T}\right)=0$, is a solution for $\beta=0$. This
corresponds to the static, non-rolling case without activity. Next, the
polynomial $p\left(\frac{t_{m}}{T}\right)$ is always negative (except for
$\frac{t_{m}}{T}=0$), with its modulus increasing with $t_{m}$. Therefore,
only when the term $\beta>0$ is present, there is the possibility to ensure
$m\propto p\left(\frac{t_{m}}{T}\right)+\frac{\beta}{\alpha+\beta}=0$, i.e.
torque balance, for finite $\frac{t_{m}}{T}$ and hence $\omega$. On the other
hand, for any $\beta>0$ there is in fact always a solution $t_{m}=t_{m}^{*}$
at which the torque vanishes, namely

$p\left(\frac{t_{m}^{*}}{T}\right)=-\frac{\beta}{\alpha+\beta}.$

We have hence shown that, at least in two-slope approximation, any finite
enzyme activity will induce motion in the simple model.

####  IV.2.3 Case of slow rolling and high enzymatic activity: three-line
approximation

Let us discuss the case sketched in Fig. 2(3). So far we had assumed that the
bound linkers cannot decay to zero in the time interval $[0,T]$. However, for
small $\omega$ and high $\beta$, the two-line approximation breaks down if the
bound linker concentration becomes negative in $[t_{0},T]$ with $\alpha
t_{m}-\beta\left(t_{0}-t_{m}\right)=0$ or

$t_{0}=\frac{\alpha+\beta}{\beta}t_{m}.$ (34)

To cover the whole range of frequencies, we generalize the profile to a Three-
Line-Approximation (3LA) by writing

$B\left(t\right)=\begin{cases}\alpha t&\text{ for $0\leq t\leq t_{m}$}\\\
\alpha t_{m}-\beta\left(t-t_{m}\right)&\text{ for $t_{m}\leq t\leq t_{0}$}\\\
0&\text{ for $t_{0}\leq t\leq T$}\end{cases}$ (35)

We simplify the evaluation of the torque integral as follows: in the limit of
small $\omega$ (i.e. large $T$) and not too high $\beta$ (such that
$t_{0}>t_{m}$ and $\frac{\alpha+\beta}{\beta}\simeq\frac{\alpha}{\beta}$) the
integral over the second region dominates over the first, which is very small
(rapid rise to the plateau), and the third region does not contribute anyways.
It is hence enough to evaluate

$m\simeq-\frac{m_{0}\omega^{4}}{H_{0}}\int_{t_{m}}^{t_{0}}\left(t-\frac{T}{2}\right)^{3}\left(\alpha
t_{m}-\beta\left(t-t_{m}\right)\right)\,dt$

Moving the lower boundary to $0$ and using
$t_{0}\simeq\frac{\alpha}{\beta}t_{m}$ (as discussed above) one gets by
neglecting terms which are a factor $t_{m}/t_{0}$ smaller

$m\simeq-\frac{m_{0}\omega^{4}}{H_{0}}\frac{\beta}{80}T^{5}p\left(\frac{t_{0}}{T}\right)$
(36)

with still the same polynomial, cf. Eq. (27).

####  IV.2.4 Full torque-angular velocity relation

Using $T=\frac{2\phi_{c}}{\omega}$ one can introduce the characteristic
frequency

$\omega_{0}=\frac{2\phi_{c}}{t_{0}}=\frac{2\phi_{c}}{t_{m}}\frac{\beta}{\alpha}$
(37)

to replace the argument of the polynomial in Eq. (36) by $\omega/\omega_{0}$.
Rescaling as $\tilde{\omega}=\omega/\omega_{0}$ one gets the following torque-
angular velocity relation for small frequencies, $\tilde{\omega}<1$, i.e.
$\omega<\omega_{0}$:

$m=m_{0}\frac{B_{pl}}{H_{0}}\frac{\phi_{c}^{4}}{5}\tilde{\omega}\cdot\left(-4\,\tilde{\omega}^{3}+10\,\tilde{\omega}^{2}-10\,\tilde{\omega}+5\right).$
(38)

For larger frequencies , $\tilde{\omega}>1$, we can use Eq. (31), which can be
rescaled the same way to yield

$m=-m_{0}\frac{B_{pl}}{H_{0}}\frac{\phi_{c}^{4}}{5}\left(5\frac{\beta}{\alpha}\tilde{\omega}-\frac{1}{\tilde{\omega}}\right).$
(39)

We hence have established the full torque-angular velocity relation for a
rolling virus. While $\alpha$ is determined simply by the HA-glycan on-
kinetics, $\beta$ is yet not specified. We will determine it in the next
section by a perturbation expansion of the non-enzymatic state and postpone
the discussion of the steady rolling and the torque-angular velocity relation
to section IV.4.

###  IV.3 Including enzymatic activity: perturbative solution

We now include the enzymatic activity of NA and treat it as a perturbation of
the forced-rolling steady state, obtained in section IV.1, where HA-glycan
binding leads to the two-line profile with a rapid increase at the front
(characterized by $\alpha$) leveling off to a plateau.

We hence assume that $\epsilon=V_{cut}/\alpha$ is a small parameter i.e. the
enzyme cutting activity is small against the binding kinetics. We then can
write

$B=B_{1}+\epsilon B_{2}+\ldots\,,\,\,G=G_{1}+\epsilon G_{2}+\ldots.$ (40)

such that the zero order $\mathcal{O}(\epsilon^{0})$ is just the passive (i.e.
forced rolling) case of section IV.1.

To next order $\mathcal{O}(\epsilon)$ one has to consider

$\displaystyle\omega B_{2}^{\prime}$ $\displaystyle=$ $\displaystyle
k_{on}H_{0}G_{2}-k_{off}B_{2}-k_{on}\left(G_{1}B_{2}+B_{1}G_{2}\right),$
$\displaystyle\omega G_{2}^{\prime}$ $\displaystyle=$ $\displaystyle-
k_{on}H_{0}G_{2}+k_{off}B_{2}+k_{on}\left(G_{1}B_{2}+B_{1}G_{2}\right)$ (41)
$\displaystyle-\frac{k_{on}H_{0}G_{0}G_{1}}{K_{M}+G_{1}}.\,\,\,\,\,\,$

As the binding kinetics is assumed to be fast, we use an adiabatic
approximation, $\omega B_{2}^{\prime}\simeq 0$. Then the equations for $G_{2}$
and $B_{2}$ simplify to

$\omega
G_{2}^{\prime}=-\frac{k_{on}H_{0}G_{0}G_{1}}{K_{M}+G_{1}}\,,\,\,\,B_{2}=\frac{H_{0}-B_{1}}{K_{d}+G_{1}}G_{2}\,.$
(42)

Neglecting the boundary layer close to the front, we can replace $B_{1}\simeq
B_{pl}$, $G_{1}\simeq G_{pl}=G_{0}-B_{pl}$ by their plateau values. Also note
that $G_{2}(t=0)=0$, since the boundary conditions at the front are already
fulfilled in zero order. Transforming to time space we get the dynamical
profile for the glycans
$G_{2}\left(t\right)=-\frac{k_{on}H_{0}G_{0}G_{pl}}{K_{M}+G_{pl}}t$ and hence
for the bound linkers

$B_{2}\left(t\right)=-\frac{H_{0}-B_{pl}}{K_{d}+G_{pl}}\frac{k_{on}H_{0}G_{0}G_{pl}}{K_{M}+G_{pl}}t\,.$
(43)

Transforming back to angle space and using $\epsilon=V_{cut}/\alpha$ where
$\alpha=k_{on}H_{0}G_{0}$, yields [25] $B_{2}=-\alpha
f\,\frac{\phi+\phi_{c}}{\omega}$ or

$\beta=V_{cut}f\,,\,\,\,f=\frac{H_{0}-B_{pl}}{K_{d}+G_{pl}}\frac{G_{pl}}{K_{M}+G_{pl}}.$
(44)

This result directly implies an expansion for the torque,

$m=m_{1}+\epsilon m_{2},$ (45)

where $m_{1}=m_{diss}$ is the passive torque, given in Eq. (28), and $\epsilon
m_{2}=m_{act}$ is the active torque. Explicitly one obtains

$m_{act}=\frac{p_{act}}{\omega}\,,\,\,p_{act}=\frac{m_{0}}{H_{0}}f\,\frac{2\phi_{c}^{5}}{5}V_{cut}\,.$
(46)

Here $p_{act}$ is the power injected by NA operation. The active torque is
positive (since $B_{pl}<H_{0}$), it is proportional to $V_{cut}$ and has a
$1/\omega$ dependence, unlike the passive one which is linear in $\omega$.

###  IV.4 Discussion of rolling velocity and force-velocity relation

Let us now discuss the results obtained analytically. In the case of weak
enzymatic activity, we can combine Eq. (28) for the dissipative torque and Eq.
(46) for the active driving torque to the torque balance
$m_{diss}+m_{act}=0=-\xi_{diss}\omega+p_{act}/\omega$. This immediately
implies a pitchfork bifurcation for the steady-state rolling velocity

$\omega=\pm\sqrt{\frac{p_{act}}{\xi_{diss}}}\propto\phi_{c}\sqrt{f}\,\frac{\sqrt{\alpha
V_{cut}}}{B_{pl}}\,.$ (47)

This is exactly what had been observed in Fig. 3a), the velocity scaling like
$\omega\propto\pm\sqrt{\beta}$ with $\beta\propto V_{cut}$ with $V_{cut}$
linear in the NA concentration. As the total torque is zero, the torque scale
$m_{0}=\frac{1}{2}SR^{2}\rho_{HA}$ cancels out. Nevertheless the parameters
$S$ (linker stiffness) and $R$ (radius of the virus) are still present since
they enter the contact interval size $\phi_{c}$. The result in fact depends on
all model parameters, especially the kinetics of attachment/detachment and
cutting contained in $f$.

Using the specific parameter values for influenza given in section II.1, Eq.
(47) yields values of the order of $\omega=0.4\,{\rm s}^{-1}$. This compares
well to the experimentally measured values by Sakai et al. [4, 5]: there,
translational speeds of of $v\simeq 10-30\,{\rm nm/s}$ were reported,
corresponding to $\omega$ between $0.2-0.6\,{\rm s}^{-1}$ for physiological NA
activity.

To critically discuss the torque-angular velocity relation, we will introduce

$A=\frac{\beta}{\alpha},$ (48)

which is an “activity parameter” proportional to the enzymatic activity.
Adding now a term $5A\tilde{\omega}$ to Eq. (38), which represents a small
correction since $\beta\ll\alpha$, allows us to combine the two obtained
limits, Eq. (38) and Eq. (39) to one continuous curve,

$\tilde{m}\left(\tilde{\omega}\right)\approx-\begin{cases}\tilde{\omega}\cdot\left(-4\tilde{\omega}^{3}+10\tilde{\omega}^{2}-10\tilde{\omega}+5\left(1-A\right)\right)&\text{;
$\tilde{\omega}$$\leq 1$}\\\ \tilde{\omega}^{-1}-5A\tilde{\omega}&\text{;
$\tilde{\omega}$$>1$}\end{cases}$ (49)

where in addition we non-dimensionalized the torque
$\tilde{m}=\frac{m}{m_{c}}$ with
$m_{c}=m_{0}\frac{B_{pl}}{H_{0}}\frac{\phi_{c}^{4}}{5}$. Note that, maybe
counterintuitively, the activity is now in front of the passive torque for the
branch $\tilde{\omega}>1$. This is due to the characteristic frequency,
$\omega_{0}=\frac{2\phi_{c}}{t_{0}}=\frac{2\phi_{c}}{t_{m}}A$, being
proportional to the activity parameter: the higher the activity, the larger
the rolling frequency must be to prevent the depletion zone at the back (the
$B=0$ region) to occur, and to stay in the regime $\tilde{\omega}>1$.

Figure 5:  Torque-velocity relation obtained from the numerical solution of
the continuum model (red curve), from the approximate multi-line theory, Eq.
(49), with $\beta$ from the perturbation theory (green curve), and from
stochastic simulations (symbols). Close to steady torque-free rolling, the
agreement between numerics and stochastic simulations is excellent; the
approximate theory displays the correct overall shape. The torque is given in
pN$/L$ where $L$ is the virus’ length along the cylinder axis. For the
stochastic simulations, the velocity has been averaged over the whole
simulated trajectory with error bars displaying standard deviation. The points
$P_{1},P_{2},P_{3}$ characterizing the ”motor curve” are discussed in the
text.

Fig. 5 shows the torque-velocity relation from a numerical solution of the
continuum model (red curve) as explained in section III.2, and the approximate
multi-line theory (green curve). The stochastic simulation results (symbols)
shown additionally are explained in the next section. In view of the
approximations made, the green curve given by Eq. (49) captures the behavior
well on the semi-quantitative level and displays the same features as the
numerically obtained red curve, namely three characteristic points:

($P_{1}$): for $\left(\omega,m\right)=\left(0,0\right)$ the virus is in the
immobile state, which always should be a solution. ($P_{2}$): the point
$\left(\omega,m\right)\approx\left(\frac{\sqrt{5}}{2}\omega_{0},\left(5.6\,A-0.8\right)m_{c}\right)$
characterizes the maximum sustainable torque (in the deterministic, mean-field
model). For counter torques of larger amplitude, the motion is no longer
stable. And finally ($P_{3}$): for
$\left(\omega,m\right)=\left(\frac{\omega_{0}}{\sqrt{5A}},0\right)\propto\left(A^{1/2},0\right)$
the system is in the steady, self-propelled rolling state and torque free, a
behavior that is well captured already by Eq. (47).

Concerning ($P_{2}$), interestingly for small activity $A\rightarrow 0$ the
maximum torque the virus can sustain becomes independent of the activity
parameter: $m\to-0.8m_{c}$. The corresponding minimal speed under subcritical
forcing vanishes linearly with activity as
$\omega=\frac{\sqrt{5}}{2}\omega_{0}\propto A\rightarrow 0$. In the same
(singular) limit, $A\rightarrow 0^{+}$, the torque-free motion, i.e. point
($P_{3}$), has no threshold and scales as $\omega_{free}\propto
A^{1/2}\propto\phi_{c}\sqrt{\alpha V_{cut}}$, as already derived previously.
Thus, in the absence of external torque and for arbitrary low activity $A$,
one always has an (arbitrary slow) self-sustained rolling motion.

##  V Stochastic simulations

###  V.1 Implementation and comparison stochastic vs. continuum model

So far we investigated the problem of rolling on the mean-field level, where
we could show the existence of a steady rolling state for a virus having
enzymatic NA activity. The questions whether this state is stable in view of
the strongly fluctuating conditions at the nano scale and how the virus
actually reaches this state, i.e. self-polarizes, demand a stochastic modeling
framework. We here briefly explain the implementation of the model using the
Gillespie algorithm (with details given in appendix C) and then critically
compare the stochastic simulations to the continuum theory.

Let us consider a cylindrical virus, cf. Fig. 1, of length $L$ and presenting
a number of $N_{vir}\propto L$ discrete binding sites and project the cylinder
onto a single cross-section. Then all binding sites (that one can assume
randomly densely packed along the virus surface) get projected onto this
circle. As the contact zone size, $2\phi_{c}R$ is fixed, the relation
$N_{vir}\Delta x=2\phi_{c}R$ defines the effective size $\Delta x$ of a
projected binding site. $N_{vir}\rightarrow\infty$ (i.e. $\Delta x\rightarrow
0$) represents the continuum deterministic limit. In turn, small $N_{vir}$ can
be interpreted as a circular virus, for which, using the size of an HA and a
circular contact area, one estimates $N_{vir}\simeq 20$.

In the numerical algorithm, space is discretized by $\Delta x$ on a large box
(having typically $N=2000\gg N_{vir}$ sites) with periodic boundary
conditions. The angular variable used in the mean field model is discretized
accordingly, i.e. by $\phi=2\phi_{c}n/N_{vir}$ defining the discrete bound
linkers and free glycans, $B=B[n]$ and $G=G[n]$, respectively. To obtain the
effective molar concentrations of the spike proteins and the ligands one needs
to convert from known surface densities on the virus and the substrate to
volume densities. Estimating the charateristic volume of a molecule by
$V_{m}\sim(10\,{\rm nm})^{3}$, the molar concentration is given by
$C_{M}=\frac{1l}{N_{A}V_{m}}\simeq 1\,{\rm mM}$, or in other words, 1
molecule$/V_{m}$ corresponds to $\simeq 1\,{\rm mM}$.

The Gillespie method [26] is an event-driven algorithm, and time
$t_{i}=\sum_{m=1}^{i}\Delta t_{m}$ is discretized in waiting times $\Delta
t_{m}$. For every waiting time $\Delta t_{m}$ one considers all possible
events, i.e. binding, unbinding and cutting with their discretized rates. The
waiting times are drawn according to $\Delta t_{m}=-\frac{\ln\xi}{a_{T}}$
where $\xi\in[0,1]$ is a uniform random variable and $a_{T}$ is the sum of the
rates of all possible events (see appendix C for details). A second random
number is used to choose which event takes place.

Finally, one has to evaluate the torque balance using the newly obtained
linker configuration to determine the new center of mass position of the
virus, $s(t_{i+1})$. Torque balance is assumed to be instantaneously
established and, expressing the contact interval $[-\phi_{c},\phi_{c}]$ via
the binding sites $[n_{L},n_{R}]$, the discretized torque balance reads

$\sum_{n=n_{L}}^{n_{R}}\left(n-s\right)^{3}\,B[n]=0.$ (50)

This is a cubic equation for $s=s(t_{i+1})$ (for details see appendix C) and
always has a real solution that can be determined by root finding after every
event that changes $B[n]$. Having the new position $s$ (the nearest binding
site is chosen), we get the new binding interval by shifting $B$ to its new
center of mass position $s$. This simulation yields trajectories $s(t_{i})$,
from which angular velocities $\omega(t_{i})$ and velocities
$v(t_{i})=R\omega(t_{i})$ can be readily determined, as well as the profiles
$B[n],G[n]$ at every $t_{i}$.

To compare the stochastic implementation to the mean-field, continuum model we
first choose a high number of linkers, $N_{vir}=200$, such that effects like
stochastic reversals of the direction of motion or even complete detachment of
the virus do not occur (see next section for a discussion of these effects).
If not stated otherwise, we simulate the system for the parameters, already
given and discussed in section II. Going back to Fig. 5 that compares the
torque-velocity relation obtained from stochastic simulations (symbols) to
those from the numerical solution (red curve) and the approximate multi-line
theory (green curve) of the mean-field model, one can see that the numerics
and the stochastic simulations agree very well in a substantial region around
the torque-free rolling.

Interestingly, in the stochastic simulations we were unable to get the full
branch from the torque-free rolling down to the minimum of $m(\omega)$: the
virus always reverted its direction for large counter-torque, i.e. the system
jumped to the negative velocity branch – note that for negative velocity, the
torque velocity is the same curve as displayed for positive velocities, but
upside-down. Without stochasticity, one would expect the branch to be stable
down to the minimum and only the branch from the minmum up to
$(m,\omega)=(0,0)$ to be unstable. The latter branch corresponds to bound
linker distributions with a three-line profile, i.e. where the glycan is
completely cut at the back. This suggests that these states/profiles are
important to understand the full torque-angular velocity relation, but that
they are dynamically unstable, especially in the presence of stochastic
fluctuations.

Figure 6: Study of a virus (radius 50 nm, $N_{vir}$ binding sites) rolling on
a substrate with periodic boundary conditions (total number of binding sites
2000). Panel a) shows the average time until the virus detaches completely
from the substrate as a function of the number of linkers $N_{vir}$. Panel b)
shows the average glycan remaining on the substrate after the virus has
detached, again as a function of the number of linkers. Averages were taken
over 20 realizations for every value of $N_{vir}$, shaded areas show the
standard deviation.

###  V.2 Stochastic effects: virus detachments and reversals

Figure 7: Study of a virus (radius 50 nm, $N_{vir}$ binding sites) rolling on
a substrate with periodic boundary conditions (total number of binding sites
2000) and infinitely rapid glycan recovery outside the virus’ contact zone.
Panel a) shows the frequency of reversals (number per time) and panel b) the
average run length between two reversals. Panel c) gives the average velocity
obtained from the two former quantities. For the blue data the glycan
distribution on the substrate was perfectly homogeneous at the standard value
used, $G_{0}=10$, while for the red data a noisy distribution was implemented
(with strong noise amplitude of $G_{0}/2$). All averages were taken over 20
runs of 500 s running time each, shaded areas show the standard deviation.

The stochastic implementation in addition allows to investigate the effects of
finite linker numbers on the overall behavior of virus rolling, inducing for
instance reversal of rolling direction or even complete detachment. Decreasing
$N_{vir}$ to values below 100 (for the given, realistic parameters), Fig. 6a)
shows the time it takes for a virus, initially placed on a homogeneous glycan-
covered substrate, to get completely detached – i.e. reaching the state with
$B[n]=0$ for all $n$ in the contact zone. In turn, Fig. 6b) shows the averaged
amount of glycan that is still left on the substrate, after the virus has
detached.

Fig. 6 suggests that the origin of detachment for small values of $N_{vir}$ is
due to intrinsic stochasticity of binding, as the glycan level after
detachment is still high. In contrast, for large values of $N_{vir}$ the virus
rather detaches since the glycan level becomes low. The latter is due to the
fact that the virus consumes part of the substrate while rolling. It may also
change direction of movement or go through the periodic box such that it
crosses again the same, already partially glycan-depleted region more than
once.

If the substrate were not consumed one would expect the time to get detached
to grow exponentially with the number of linkers, since so does the number of
possible configurations of $B[n]$. In order to study reversals and run times
in a way that is not influenced by the history of the virus’ path, in the
following we will consider the case were the glycans are recovered by the
cell. For simplicity we assume this recovery to be infinitely fast.
Specifically, whenever a virus has rolled over a part of the substrate, the
respective value $G(n)$ is restored to the initial level $G_{0}$. This
corresponds to a virus rolling on a cell that has a high membrane diffusivity
and rapidly reshuffles its surface glycans. We also come back to this point in
section VII; note that glycan reshuffling via lateral diffusion within the
cell membrane has been readily observed in experiments [27, 28].

Implementing this infinitely fast glycan recovery outside of the contact zone
of the virus one can study the statistitics of the reversals of direction of
the rolling virus and quantify the typical runlengths as a function of
parameters and independent of the simulation box size. The results are shown
in Fig. 7. The blue data in Panel a) show the frequency of reversals as a
function of the number of linkers for homogeneous glycan distribution $G_{0}$
outside the contact zone. For the red data, we added a stochastic glycan
distribution (noise level $50\%$ of mean value). In both cases, the reversals
decrease if the linker number increases. For the blue data, these reversals
are solely due to the intrinsic stochasticity of the dynamics. As expected,
for the noisy glycan distribution, reversals occur more frequently, but the
rolling is still robust.

Fig. 7b) shows the average runlengths, i.e. the distance traveled between two
reversals, obtained from the same raw data. The runlength increases with the
number of linkers and can easily reach several microns (meaning many tens the
virus diameters) for a homogeneous glycan distribution. Noisy glycan
distributions impede this substantially, but the virus still travels few times
its size for still moderate linker numbers. Finally, from panels a) and b) one
can estimate the mean velocity of the virus as shown in Fig. 7c). This is in
good qualitative agreement with the theory/continuum model, which for the
given parameters (and $N_{vir}\rightarrow\infty$) is 20 nm/s, cf. section
IV.4.

##  VI Further topics

###  VI.1 Approximations made, especially detailed balance for on-off kinetics

Within the simple model approach it is very satisfying to see the good
agreement between the numerical and analytical approaches to the mean field
model and the stochastic implementation. However, to be amenable to an
analytical treatment, we made several – in part strong – approximations that
should be critically discussed.

First, in section II.3 concerning the torque balance, we applied a small angle
approximation for the contact angle. We checked numerically that for the
given, realistic parameters, this has only a minor quantitative effect.
Second, we neglected the effect of linker compression. In small angle
approximation, linker compression can be included even analytically, with
details given in appendix B. Again, this only leads to a quantitative
correction.

The most critical approximation was made in section II.4: namely, the simple
model only approximately fulfills detailed balance for the HA-glycan on-off
kinetics. In fact, while in section II.2 we used the force-dependence (and
hence angle-dependence) of the attachment/detachment kinetics of the linkers
to determine the size of the contact interval, this dependence was neglected
in the dynamic equations, Eqs. (7), (8).

To improve on this point we now assume that the dissociation constant
increases with the “Boltzmann factor” of the elastic stretch energy

$\displaystyle\frac{K_{d}(\phi)}{K_{d}(0)}=\exp\left(\frac{E_{el}}{k_{B}T}\right)=\exp\left(\frac{SR^{2}}{8k_{B}T}\phi^{4}\right).$
(51)

The equilibrium probablity distribution of the bound linkers is then given by
$B_{pl}(\phi)=\frac{C_{0}-C_{1}}{2}$, as given in section IV.1, but now with
the angle-dependent $K_{d}$ entering $C_{0}$, $C_{1}$, explicitly

$\displaystyle\frac{B_{eq}(\phi)}{H_{0}}=\frac{1}{2}\left(\bar{B}-\sqrt{\bar{B}^{2}-4\frac{G_{0}}{H_{0}}}\right)$
(52)

with $\bar{B}=1+\frac{G_{0}+K_{d}(\phi)}{H_{0}}$. The resulting $B_{eq}(\phi)$
implies a bell-shaped linker profile, with a maximum at $\phi=0$ and decaying
rapidly (like $\exp(-\phi^{4})$) for finite angles.

We can now determine the size of the contact interval more properly: defining
$\phi_{c}$ as the point where $p_{\rm bound}=\frac{B_{eq}(\phi)}{H_{0}}$ drops
below $1/2$, from Eq. (52) one gets $G_{0}-\frac{H_{0}}{2}=K_{d}(\phi_{c})$.
For typical parameters, $H_{0}/2\ll G_{0}$ holds. Inserting Eq. (51) and
solving for $\phi_{c}$ then yields exactly the scaling result, Eq. (3).

Figure 8:  Test of the effect of detailed balance of the on-off kinetics.
Shown are bound linker distributions obtained numerically (solid) and by
stochastic simulations (symbols). The blue curves are for the simple model
(discussed throughout so far, without detailed balance), while the green
curves obey detailed balance, i.e. $K_{d}(\phi)$ is angle-dependent as given
by Eq. (51). In the latter case, the sloped linker profile in the center
prevails and the virus is still able to roll steadily. Stochastic simulations
are averaged over $10^{6}$ realizations.

To scrutinize the effect of the detailed balance of the on-off kinetics on the
bound linker distribution and ultimately on the rolling motion, we compared
numerical solutions of the mean field model to stochastic simualtions, both
without detailed balance (constant $K_{d}$) and with detailed balance, i.e.
$K_{d}(\phi)$. The result is shown in Fig. 8. The blue curve shows the case
studied before, the two slopes quantified by $\alpha$ and $\beta$ being
clearly visible. The green curve shows the case with detailed balance. One can
see that the bound linker profile decays much stronger both in rolling
direction (to the left) and at the trailing edge (to the right), but overall
the sloped distribution in the center region prevails and is sufficient to
allow persistent rolling motion. We also checked that the speed is only
quantitatively affected (for the given, realistic parameters by 10-20%).

###  VI.2 Linker distribution as an ”internal flywheel”

As observed already in Ref. [11], the Gillespie simulation-based computer
experiments reveal a surprising robustness of the rolling virus: even when
facing obstacles in form of glycan-depleted spots on the surface, the virus
often just rolls over them as if it possesses an internal ”inertia”. A once
directionally polarized virus can even persistently roll against a glycan
gradient as long as there is a sufficient amount of glycan left to bind. Thus
the virus is not performing a (chemo)taxis on the glycan concentration (as one
would expect for a burned bridge Brownian ratchet). Instead the glycan acts
here as a mechano-chemical free-energy source in analogy to the role of ATP
for classical molecular motors. The fact that the glycan is confined (or
sometimes even immobilized) on a 2D surface while ATP typically freely
diffuses in 3D is only a superficial difference.

To understand the origin of the observed processivity behavior , we pose the
following question: What happens when a steady rolling virus at initial
angular velocity $\omega_{free}$ is suddenly stopped – by an external force,
such as an immovable obstacle – at time $t=0$? How does its torque $m(t)$
dynamically respond to this sudden constraint?

Figure 9:  Demonstration of the “mechano-chemical flywheel” effect. A virus
that was rolling steadily was suddenly stopped at $t=0$. Since the internal
linker profile needs time to relax, a characteristic buildup of torque occurs
with a characteristic time of the order of $10\,$s. Shown are results from
stochastic simulations (blue; averaged over 10 events) and the scaling
estimate, Eq. (53), in red (with no fitting parameter). Linker stiffness
$S=0.1k_{B}T/{\rm nm}^{2}$, other parameters as given in section II.

Initially, for $t<0$, the virus was rolling in a torque-free state, implying
that any torque buildup starts from $m(0)=0$. The virus being blocked for
$t>0$, the advective (rolling) term is now missing in Eqs. (7), (8) and the
linker distribution in the front region will respond by rapidly equilibrating
to the plateau-value there. This front equilibration has the timscale $t\sim
B_{pl}/\alpha$ and due to torque imbalance it is accompanied by an buildup of
torque $m(t)$. While the front region’s slope (cf. Fig. 2) becomes flatter
during this fast and transient process, the more extended, cutting-dominated
rear region stays roughly unperturbed as it responds much slower. This implies
a maximum torque $m_{max}\propto-\beta/\omega_{free}$ given only by the
cutting-induced gradient at the rear. Finally, on the long timescale $t\sim
B_{pl}/\beta$, the progressive NA linker-cutting makes the linker profile
decay to zero in the whole contact zone, leading to a gradually vanishing
torque $m(t)\rightarrow 0$. In summary, a suddenly stopped virus responds with
a transient torque buildup for $t>0$ of the approximate form

$m(t)\simeq
m_{max}\tanh\left(\frac{\alpha}{B_{pl}}t\right)e^{-\frac{\beta}{B_{pl}}t}$
(53)

where $m_{max}\simeq p_{act}/\omega_{free}$.

Figure 9 shows a comparison of the torque buildup measured in stochastic
simulations (averaged over 10 stopping events) and the scaling formula, Eq.
(53). It confirms the interpretation that a stopped virus mobilizes its bound
linkers and puts up dynamic resistance against the obstacle. One could say
that the system formally behaves as if it possessed a built-in ”flywheel” –
here of mechano-chemical origin – that is dissipatively coupled to the rolling
angle variable and tends to maintain its angular momentum. This ”flywheel”
dynamic response is associated to the re-equilibration of an internal
nonequilibrium steady state. It seems to be a defining feature of other
dissipative rolling objects as well [45, 35, 36] and deserves closer
inspection in the future.

##  VII Discussion

We have proposed and analyzed a new mechanism allowing a virus to actively
roll along a substrate and shown that this rolling is essentially inevitable,
once few basic conditions are met: namely, (1) existence of a rolling axis for
the virus, (2) presence of ligands on the substrate, and (3) viral spikes (one
or several types) that bind and enzymatically cut these ligands. The obvious
question to ask next is: how does the virus benefit from it?

### Why the rolling?

Obviously, being a nanoscale object the virus could just move through the bulk
of a low viscosity fluid via thermal diffusion. However, in the case of mucus-
binding viruses like influenza, the large viscosity and the gel-like nature of
the environment severely limits this. An additional danger influenza faces is
getting stuck in/to the mucus and eventually being swept away by the beating
cilia of our respiratory tract. To make the situation even worse for the
virus, during its entry phase [29, 30] the virus can become localized at wrong
spots on the cell membrane (or on a wrong cellular structure like a cilium),
being unable to enter the cell at all. If the cell membrane is sufficiently
fluid, two-dimensional (2D) diffusion can help out, but not if the glycan
receptors (glyco-lipids and proteins) are trapped within local membrane
domains controlled by the cytoskeleton underneath. Influenza could respond by
weakening its bonds with the immobile glycans in order to still diffuse on the
quasi-rigid substrate, the drawback being that increasing diffusivity also
increases the virus-membrane detachment rate. All these are likely reasons for
influenza having evolutionary come up with the active enzyme solution.

In spite of sufficiently strong collective, multi-linker binding by the HA-
spikes, the virus can still actively weaken these links via the NA-spikes’
cutting activity. However, if this form of ”stick-and-cut” behavior was
spatially uncoordinated, it would be difficult to comprehend how it could give
rise to any sufficiently fast or even directed motion. How should the virus
decide which way to go and keep a certain persistence of direction? Obviously,
the (immobile) glycan-covered substrate could help the virus to some extent by
keeping records of where the virus has already been: the virus would then
simply statistically avoid the NA-generated glycan-depleted regions and
perform a form of 2D self-avoiding walk, which is the idea underlying the so-
called burnt-bridge Brownian ratchet [9, 10]. This strategy could be enhanced
by a polarized distribution of the spikes [20], with NA enriched at the rear,
depleting more glycans there, and HA enriched in front, both effects enhancing
forward binding and stabilizing directional movement. Indeed this is observed
for IVA where HA and NA are physically separated molecules “floating” rather
freely in the membrane, where they can polarize the virus via a partial phase
separation. The motion in this case happens along the long axis of the
ellipsoidal or filamentous virus. However, for IVC the two spikes are “glued
together” into a single, inseparable unit, the HEF protein [19]. Naively, this
additional constraint would make the virus less motile, yet the opposite is
true: IVC moves about 5-10 times faster than IVA [4, 5]. In addition to its
larger speed, most notably IVC moves orthogonally to its long axis.

It seems that these two propulsion modes – parallel and orthogonal to the axis
– have radically different physical mechanisms. Here we suggested that the
motion orthogonal to the cylindrical axis is tightly coupled to axial
rotation. This coupling is more than just an easy, low dissipation mode of
motion, but rather intrinsically linked to the very mechanism of dynamic
linker polarization: on the one hand, the angular rotation itself leads to a
linker polarization within the virus-substrate contact zone. On the other
hand, the linker polarization front vs. rear gives rise to a torque and the
angular velocity. Both interdependent effects are inseparable and make rolling
propulsion fast, robust and efficient compared to longitudinal gliding.

The emergence of rolling can be seen as a step in the evolutionary race
between the virus and the host. Naturally, this race is still ongoing and the
host could also take counter measures. One possibility is the mixing-up and
quick replacement of the cut glycans by fresh ones (cf. the glycan recovery in
section V.2). However, for this to be efficient it has to affect the contact
zone, as the rolling virus is not very sensitive to gradients outside. Hence
glycan replacement must be extremely fast (the virus passes the contact zone
within few seconds). Another possibilty is to ”clog” the rotation by firmly
binding ligands (antibodies) to some of the free HA/HEF. In this case, the
virus would have to resort to other, less efficient propulsion mechanisms like
gliding. Alternatively, it could counteract the clogging by allowing
sterically blocked HA/HEF to float by keeping the transmembrane proteins
(including the spikes) in a fluid state. As just sketched here, the game-
theory of this intricate evolutionary race promises new surprises for future
studies.

###  Experimental Questions

From the experimental point of view, advancing elegant in vitro setups like
those by Sakai et al. [5, 4] should allow to quantitatively probe the detailed
mechanism proposed here, the most robust and easy-to-test predictions being:
(1) The rolling speed as a function of NA, HA, and glycan surface-
concentrations. Especially the characteristic square root relation
$\omega\propto\sqrt{V_{cut}}$, cf. Eq. (47), between the angular velocity and
the cutting rate $V_{cut}\propto[NA]$.
(2) The force-velocity or “motor relation” shown in Fig. 5 and the
characteristic points $P_{1}$-$P_{3}$ therein, as well as the scaling behavior
of the curve.
(3) The instantaneous force response of a stalled, immobilized virus, as shown
in Fig. 9 and quantified in Eq. (53).

Experimentally these should be accessible as the filamentous viruses are large
(long axis one micron) and sufficiently slow (tens of nm/s) to be readily
captured by various microscopy methods. Even smaller viruses have been studied
already using high resolution techniques [31]. The force magnitudes (tens of
pN) and their moderately slow time evolution (few seconds) are also well
within the range of common force spectroscopy methods [32, 33]. To test the
theory it is most practical to utilize filamentous viruses. Although the model
developed here also applies to spherical viruses on small time scales, the
high rotational diffusion constant of a tiny nanosphere and the small number
of linkers making contact ($N_{vir}\simeq 20$) will in this case give rise to
large orientational and velocity fluctuations. As for any directionally self-
propelled object of characteristic size $L$, there will be a crossover from
ballistic to diffusive motion at a timescale $t_{rot}\propto 1/D_{rot}$ with
$D_{rot}\propto L^{-3}$ the rotational diffusion constant. While for a sphere
$L$ will be the radius, for a cylinder it will be its length, making the
directed propulsion of the latter experimentally much easier to probe.

Many interesting questions still remain, including the role of the different
motility modes – gliding by spike polarization [20] and rolling [5, 4] – under
similar conditions. Can IVA actually switch between the two motility modes?
And how do IVs finally switch from rolling to entering the cell [29, 30, 34]?

On the physics side there is an interesting conceptual question that still
needs to be addressed: Other non-equilibrium filamentous rollers are known to
exist, but these are bulk-driven, meaning there is a matter-energy flow (e.g.
of heat [35] or solvent [36]) through the cross-section of the cylinder. So
what are similarities and differences of active rollers that are bulk-driven
vs. surface-driven in view of a general description, their efficiency, or the
existence and nature of their internal flywheel modes?

##  VIII Conclusions

In conclusion, the rolling mechanism renders influenza an even smarter
adversary than previously thought. In contrast to classical virology dogmas,
this virus type displays a “surface metabolism” harvesting the chemical energy
of sugars on the host’s membrane for directional force generation. In turn,
this transforms the whole virus capsid into a complex rotary motor. The
underlying linker dynamics shares similarities with passive (e.g. shear
induced [37, 38, 39]) or other [40] adhesive rolling mechanisms, as well as
with collective motor ensembles [41, 42, 43], but with the crucial addition of
the enzymatic substrate cutting. The mechanism discussed here for influenza
should also apply to other viruses having enzymatic spike proteins: candidates
are the toro-virus and some of the beta-corona-viruses [44]. We can also flip
the coin and learn from the virus’ workings: in fact, the combination of a
binding molecule, a cutting enzyme and the spherical/cylindrical geometry has
been already used to propel DNA-coated beads [45, 46, 47] along RNA-covered
surfaces and the insights developed here could now be used to optimize such
and related artificial rollers.

##  IX Appendix

##  Appendix A Calculation in angle space and scaling

The calculation of the torque becomes especially transparent in angle space,
where the two-line linker distribution reads

$B\left(\phi\right)=\begin{cases}\frac{\alpha}{\omega}(\phi+\phi_{c})&\text{
for $-\phi_{c}\leq\phi\leq\phi_{pl}$}\\\
\frac{\alpha}{\omega}(\phi_{pl}+\phi_{c})-\frac{\beta}{\omega}\left(\phi-\phi_{pl}\right)&\text{
for $\phi_{pl}\leq\phi\leq\phi_{c}$}\end{cases}$ (54)

Rewriting the torque, Eq. (4), by again using that a constant linker
distribution is torque-free, yields

$m=-\frac{m_{0}}{H_{0}}\frac{1}{\omega}\left[\alpha\int_{-\phi_{c}}^{\phi_{pl}}(\phi-\phi_{pl})\phi^{3}d\phi-\beta\int_{\phi_{pl}}^{\phi_{c}}(\phi-\phi_{pl})\phi^{3}d\phi\right]$
(55)

In the first integral, one substitutes $\tilde{\phi}=\phi+\phi_{c}$ and uses
that $\phi_{pl}+\phi_{c}\ll 1$. In the second one uses
$\phi_{pl}\simeq-\phi_{c}$. Both amount to the same statement, that the
plateau is rapidly reached (i.e. closeby the boundary in rolling direction).
One obtains

$m=-\frac{m_{0}}{H_{0}}\left[\frac{\alpha}{\omega}\frac{1}{2}\left(\phi_{pl}+\phi_{c}\right)^{2}\phi_{c}^{3}-\frac{\beta}{\omega}\frac{2}{5}\phi_{c}^{5}\right],$
(56)

and using $\phi_{pl}+\phi_{c}=\omega t_{m}$ one regains

$m=-\frac{m_{0}}{H_{0}}\left[\alpha\omega\frac{1}{2}\left(t_{m}\right)^{2}\phi_{c}^{3}-\frac{\beta}{\omega}\frac{2}{5}\phi_{c}^{5}\right].$
(57)

As stated already in section III.1 when discussing the scaling, if the plateau
is reached sufficiently rapidly, the torque integral over the negative slope
region is independent of the dynamics. Basically, there is degradation
everywhere, resulting just in a power of $\phi_{c}$, the contact area size. It
only remains the $1/\omega$-dependence from the slope in angle space. In
contrast, for the first, passive contribution, cf. Eq. (56), the dynamics is
important: the angle-integrated slope yields a square-dependence in the
plateau angle and hence in $\omega$, which together with the $1/\omega$ from
the slope yields a linear friction $\propto\omega$.

##  Appendix B Effect of linker compression

In section II.3 we introduced a distribution of linker extensions, that was
assumed to be zero at the symmetry axis and growing quadratically with angle
$\phi$. This implied only tensile linker forces for $\phi\neq 0$ and a delta-
peak counter-force with opposite sign at $\phi=0$. Physically, this assumption
means that the stiff viral spike proteins can sustain a compressive load with
little compliance but they easily stretch when tension is applied.

One can take the concept of linear linker elasticity more seriously and allow
the virus to push down on and compress the linkers. Assuming a similar chain
stretching as before, including the possibility that the virus cross-section
is shifted downwards by a (to be determined) length $l_{0}$, one has
$l(\phi)=R(1-\cos(\phi))-l_{0}$. The energy of the elastic spring foundation
reads (for small $\phi$)

$\displaystyle
e_{el}(\phi)=\frac{S}{2}l^{2}(\phi)\simeq\frac{SR^{2}}{2}\left(\phi^{2}/2-\varepsilon_{0}\right)^{2},$
(58)

with $\varepsilon_{0}=l_{0}/R$ the vertical deformation. It is determined to
be $\varepsilon_{0}=\frac{1}{6}\phi_{c}^{2}$ by the vanishing force condition,
$\int_{-\phi_{c}}^{\phi_{c}}f_{el}(\phi)d\phi=0$, with $f_{el}=-\frac{\partial
e_{el}(\phi)}{\partial l}$. The torque balance, Eq. (4), then generalizes to

$\displaystyle m_{{\rm
incl\,compr}}=-\frac{m_{0}}{H_{0}}\int_{-\phi_{c}}^{\phi_{c}}B(\phi)\left(\phi^{3}-2\varepsilon_{0}\phi\right)\,d\phi\,.$
(59)

In addition to the third moment, there now also is a contribution from the
first moment, having opposite sign. Evaluating this torque using Eq. (54)
leads to $\omega_{{\rm incl\,compr}}=\sqrt{2/3}\,\omega$ implying a reduction
of $\simeq 20\%$.

##  Appendix C Details for the stochastic implementation

The Gillespie algorithm [26] is event driven: at any time step, one decides
whether at the picked position $n$ a binding event ($B[n]\rightarrow B[n]+1$
and $G[n]\rightarrow G[n]-1$), an unbinding event ($B[n]\rightarrow B[n]-1$
and $G[n]\rightarrow G[n]+1$) or a cutting event ($G[n]\rightarrow G[n]-1$)
occurs. The weights of these events can be read off directly from Eqs. (7),
(8):

$\displaystyle a_{b}[n]=k_{on}G[n](H_{0}-B[n])\,,\,\,\,a_{u}[n]=k_{off}B[n],$
$\displaystyle a_{c}[n]=V_{cut}\frac{G[n]}{G[n]+K_{M}},$ (60)

which must be normalized by the sum
$a_{T}=\sum_{n}(a_{b}[n]+a_{u}[n]+a_{c}[n])$ to assign probabilities to each
process. The waiting times $\Delta t_{m}$ between two events can then be
picked as $\Delta t_{m}=-\frac{\ln\xi}{a_{T}}$, with $\xi$ a uniform random
variable on $[0,1]$.

Finally, to find the new center of mass position of the virus, at each step
one solves Eq. (50), i.e. the cubic equation,

$s^{3}+c_{2}s^{2}+c_{1}s+c_{0}=0,$ (61)

with coefficients determined by the bound linkers as

$\displaystyle
c_{2}=-\frac{3}{\Sigma}\sum_{i=i_{L}}^{i_{R}}n\,B[n]\,,\,\,\,c_{1}=\frac{3}{\Sigma}\sum_{i=i_{L}}^{i_{R}}n^{2}\,B[n]\,,$
$\displaystyle c_{0}=-\frac{1}{\Sigma}\sum_{i=i_{L}}^{i_{R}}n^{3}\,B[n]\,,$
(62)

where $\Sigma=\sum_{i=i_{L}}^{i_{R}}B[n]$.

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