# Coiling of cellular protrusions around extracellular fibers

Raj Kumar Sadhu1, Christian Hernandez-Padilla2, Yael Eshed Eisenbach3, Lixia
Zhang4, Harshad D Vishwasrao4, Bahareh Behkam2, Hari Shroff4,5, Aleš Iglič6,
Elior Peles3, Amrinder S. Nain2 and Nir S Gov1 1Department of Chemical and
Biological Physics, Weizmann Institute of Science, Rehovot 7610001, Israel;
2Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061,
USA; 3Department of Molecular Cell Biology, Weizmann Institute of Science,
Rehovot 7610001, Israel; 4Advanced Imaging and Microscopy Resource, National
Institutes of Health, Bethesda, Maryland, USA; 5Laboratory of High Resolution
Optical Imaging, National Institute of Biomedical Imaging and Bioengineering,
National Institutes of Health, Bethesda, Maryland, USA; 6Laboratory of
Physics, Faculty of Electrical Engineering, University of Ljubljana,
Ljubljana, Slovenia

###### Abstract

Protrusions at the leading-edge of a cell play an important role in sensing
the extracellular cues, during cellular spreading and motility. Recent studies
provided indications that these protrusions wrap (coil) around the extra-
cellular fibers. The details of this coiling process, and the mechanisms that
drive it, are not well understood. We present a combined theoretical and
experimental study of the coiling of cellular protrusions on fibers of
different geometry. Our theoretical model describes membrane protrusions that
are produced by curved membrane proteins that recruit the protrusive forces of
actin polymerization, and identifies the role of bending and adhesion energies
in orienting the leading-edges of the protrusions along the azimuthal
(coiling) direction. Our model predicts that the cell’s leading-edge coils on
round fibers, but the coiling ceases for a fiber of elliptical (flat) cross-
section. These predictions are verified by 3D visualization and quantitation
of coiling on suspended fibers using Dual-View light-sheet microscopy
(diSPIM). Overall, we provide a theoretical framework supported by high
spatiotemporal resolution experiments capable of resolving coiling of cellular
protrusions around extracellular fibers of varying diameters.

##  I Introduction

Cellular protrusions play important roles in exploring and sensing the
extracellular environment, during cell spreading and adhesion, cell migration,
and cell-cell interaction review1 ; review2 ; review3 ; Carlson2009 ;
Veranic2008 . Lamellipodia and filopodia are protrusive structures formed at
the leading-edge of a migratory cell RIDLEY2011 ; Friedl2009 ; EILKEN2010 ;
Charras2008 ; Buccione2009 ; diz-munoj2010 . These protrusions enable cells to
adhere and spread on fiber-like surfaces Sebastien2020 ; Richard2019 ;
koons2017 , such as the fibers of the extra-cellular matrix (ECM) CLARK1982264
; ushiki2002 , as well as cylindrical protrusions of other cells, such as
glial cells spreading over neighboring axonal extensions Christine2019 . In
vitro studies of the cellular spreading and migration on fibers Sebastien2020
; Richard2019  have shown how different cell types organize on these fibers
Nathan2017 ; Svitkina1995 ; Lee2012 ; Hwang2009 ; Werner2018 ; Werner2019 ,
with the cellular shape and motility found to depend on the curvature
(diameter) of the fibers nain2014 ; KENNEDY201741 ; Nathan2017 ; Lee2012 ;
Hwang2009 ; Werner2018 ; Werner2019 .

Figure 1: Coiling events occurring at the leading-edges of cellular
protrusions while suspended on a single fiber of $200nm$ diameter. (A) 3D
render of a C2C12 cell captured with Dual-View Light Sheet microscopy. C2C12
cells were tagged with GFP actin (shown in Magenta), and the polystyrene fiber
was coated with Rhodamine Fibronectin (shown in Cyan). Scale bar is 10 $\mu$m
(B) I-IV time-lapse sequence images of cellular protrusion highlighted by the
red oval in A. Images track with arrows (frames II-IV) the path of the
membrane ruffles marked with dots in the frame I, during 10-second intervals.
(C) I-IV time-lapse sequence images that follow the same description as in B,
but for the leading-edge highlighted by the yellow oval in A. The scale bar
shown in frame I is 5 $\mu m$.

Experiments studying the membrane dynamics at the leading-edge of cellular
protrusions, have found indications for coiling (wrapping) dynamics around the
extracellular fibers. In mukherjee2019cancer ; nain2022 , protrusions of
metastatic cancer cells (breast and ovarian) were observed to coil and rotate
around the fiber’s axis in a curvature-dependent manner, while in Nils2015
similar coiling dynamics of leading-edge ‘fin’-like protrusions were observed
for several cell types (fibroblasts, epithelial, endothelial). These
protrusions are important during the cell’s adhesion and spreading, and play
an important role in maintaining the cell polarity and migration.

The mechanism that gives rise to the rotational spreading of the membrane on
the fibers is not understood at present. Here we use a theoretical model
Sadhu2021  where the leading-edge protrusion forms due to forces produced by
curved actin nucleators, combined with adhesion to the substrate. Curved
membrane proteins that recruit the actin cytoskeleton have been recently
located at the leading edge of lamellipodia protrusions Orion2021 ; Galic2019
. We show here that this model spontaneously gives rise to the coiling motion
of the membrane when the vesicle spreads over a cylindrical surface that
represents the external fiber, and therefore may offer a mechanism for the
observed coiling dynamics in cells. Motivated by this theoretical result, we
use Dual-View light-sheet microscopy (diSPIM) Wu2013 ; Kumar2014  to obtain
high-resolution imaging of these coiling protrusions (Fig.1, Movie-S1-S3),
which we compare to the theoretical predictions.

##  II Results

Figure 2: Phase-diagram of the steady-state vesicle shapes in the $R-\rho$
plane, where $R$ is the fiber radius and $\rho$ is the density of curved
proteins. The snapshots are shown for $R=4,6,8,10$ (in units of $l_{min}$),
and $\rho=3.2\%,6.4\%$ and $9.6\%$. The red regions in the snapshots denote
the curved proteins while the blue regions denote the bare membrane. Here, we
use adhesion strength $(E_{ad})=1.0k_{B}T$, $F=2.0k_{B}T/l_{min}$. (a) The
background color is showing the adhered area fraction of the vesicle, which is
maximal for large $R$ and $\rho$ (phase III). (b) The background color is
showing the variance of the vertices locations along the cylinder axis (scaled
by $10^{3}~{}l_{min}^{2}$), which is maximal for small $R$ and $\rho$, where
the vesicle is aligned along the axis (phase I). (c) The background color is
showing the angular variance of the vertices along the circumferential
direction, which is maximal when the vesicle coils over the cylinder for large
$R$ and small $\rho$ (phase IV). (d) Comparison between the transition lines
between the different vesicle shapes from the simulations (dashed lines) and
the analytical predictions (solid lines).

In our theoretical model, we consider a three-dimensional cell-like vesicle,
which is described by a closed surface, having $N$ vertices, each of them
connected to its neighbors with bonds, forming a dynamically triangulated,
self-avoiding network, with the topology of a sphere miha2019 ; Sadhu2021 ;
Sadhu_phagocytosis . The membrane contains proteins with convex spontaneous
curvature, that diffuse on the surface of the vesicle, having attractive
nearest neighbour interactions with each other. Each curved protein exerts a
force ($F$) in the direction of the local outward normal of the surface,
representing the protrusive force due to actin polymerization (see the Methods
section for details).

###  II.1 Shapes of vesicles spreading on the fiber

We start by analyzing the dynamics of how a vesicle spreads on an adhesive
fiber of a circular cross-section. Spreading of the vesicle over an adhesive
cylindrical surface is determined by the balance between the bending and
adhesion energies Sadhu2021 . The curved membrane proteins, even when passive
(do not recruit the protrusive forces due to actin polymerization, $F=0$), can
enhance the spreading by reducing the bending energy cost. This is shown in SI
sec. S1, Figs. S1, S2 (Movie-S4-S8), with a monotonously increasing adhered
area as the adhesion energy, radius of the cylindrical fiber ($R$) and the
average density of curved proteins ($\rho$) increase. These systems do not
exhibit any tendency for rotations or coiling dynamics.

In Fig.2 we describe the steady-state shapes for vesicles with active curved
proteins ($F\neq 0$), as a function of $R$ and $\rho$. In Fig. 2(a), we plot
the adhered fraction of the vesicle (background color), along with the steady-
state snapshots. We notice that there are several distinct phases of adhered
vesicles on the fiber, which are marked by the colored “transition” lines
(Fig.2d). The details of the analytic calculation of the transition lines are
given in the SI Sec. S2.

I) For small $R$ and $\rho$, the vesicle shape is elongated (aligned axially,
along the long axis of the fiber), with a very small adhered area. Since $R$
is small, and there are not enough curved proteins to form a ring-like
aggregate around the whole rim of the vesicle (as in phase II, above), the
vesicle can only adhere axially and in this way minimize its bending energy.
We identify this phase as it maximizes the axial elongation of the adhered
vesicle (Fig.2b, Movie-S9) as measured by the variance in the distribution of
vertices along the axial direction (Z-axis) (background color in Fig. 2b, and
SI sec. S3, Fig. S4).

II) For small $R$, if we increase $\rho$, the vesicle will eventually form a
flat pancake-like shape with all the proteins clustered as a ring around the
rim. The protrusive forces in this case act side-way around the ring and make
this pancake-like shape stable. These type of shapes were also observed for a
free vesicle (without any adhesive substrate) miha2019 , when the protein
density $\rho$ reaches a critical value. For small $R$ the bending energy cost
of pancake-like vesicle for wrapping around the fiber is too high, and it
remains “hanging” on the fiber koons2017  (Movie-S10).

III) At large $\rho$ (such that the vesicle is able to form a pancake-like
shape), if we increase $R$, the vesicle will fully adhere, as the adhesion
energy gain now dominates over the bending energy cost of wrapping around the
fiber. This phase is identified by having the maximal adhered area fraction of
$\sim 0.50$ (background color in Fig.2a, Movie-S11).

IV) At large $R$, if $\rho$ is small enough so that the proteins can not form
a circular aggregate around the flat (pancake-like) vesicle, they form a two-
arc phase miha2019 ; Sadhu2021 , where two aggregates of proteins form at
opposing ends of the cell, stretching the elongated membrane between them.
This shape spontaneously orients to point along the circumferential direction,
and pull the vesicle into a coiled helical structure. This phase is therefore
identified by the large overall angular spread of the vesicle along its length
(Fig.2c, Movie-S12), quantified by the variance in the angular distribution of
the vertices along the circumferential direction (background color in Fig.
2(c) and SI sec. S3, Fig. S5).

Note that in the parameter regime of phase IV, the vesicles can also form a
phenotype where all the proteins are aggregated in a single cluster. These
vesicles become motile, as observed on a flat adhesive substrate Sadhu2021 .

###  II.2 The mechanism driving the coiling phase

We now analyze the coiling phase (IV, Fig. 2), to expose the mechanism that
drives the circumferential orientation of the leading-edges of the membrane
protrusions.

We consider a two-arc vesicle, generated on a flat substrate, and place it on
the cylindrical fiber along the axial orientation as shown in Fig. 3(a). We
find that the vesicle is unstable in the axially aligned state, and the two
leading-edges spontaneously rotate to circumferential alignment, thus causing
the vesicle to coil (Movie-S13). We follow the evolution of the energy
components during this process (Fig. 3(b-e)). Globally, the adhesion energy
($W_{A}$) decreases (smaller in absolute value) at the early stage (Fig. 3(b),
time $\lesssim 200$), as the tubular middle part of the vesicle partially
detaches from the curved fiber surface and is stretched by the active force
(Fig. 3(d), inset). The overall bending energy ($W_{b}$) increases throughout
the process, as the tubular part is stretched to become thinner and is bent
(coiled) circumferentially around the fiber (Fig.3c).

Figure 3: Transition of a two arc shape from axial to circumferential
(coiling) orientation. (a) Configurations of the vesicle at different times
(in Monte-Carlo units). The red arrows on the 3rd inset are showing the
direction of active forces acting on each of the two arcs of the elongated
vesicle. (b) Variation of the total adhesion energy ($W_{A}$) with time,
showing a non-monotonic variation. (c) The total bending energy ($W_{b}$) of
the vesicle as a function of time, which is increasing monotonously during the
coiling process. (d) Total work done by the active force ($W_{F}$), increasing
the vesicle length over time (shown in the inset). The length is measured by
assuming that the vesicle is organized like a helix, with the end positions
being the centers of mass of the proteins in each of the two leading-edge
arcs. (e) The total effective energy ($W_{total}$), sum of b-d, which is a
decreasing function of time. (f) Configurations showing the edge region, as
defined by the grid-plane, and the grid-sphere at later times when the shape
is highly coiled. (g) The angle ($\theta$) between the direction of the total
force acting at the cell edge (as denoted in (f)), with the cylindrical axis
as a function of time. The inset is showing the total number of vertices
within this region. (h) Adhesion energy and (i) bending energy per vertex for
the leading-edge region (f). (j) The configuration of an arc at the leading-
edge when it is oriented axially (as in $t=0$ in (a)), versus
circumferentially (as in $t=600$ in (a)). The arrows denote the active axial
stretching when the leading-edge is oriented circumferentially, increasing the
adhesion energy (h). Here, we use $E_{ad}=2.0k_{B}T$, $\rho=2.4\%$ and
$F=2.0k_{B}T/l_{min}$. The unit of time in the plots is normalized to be
$2\times 10^{4}$ MC steps.

These energy penalties are counter-balanced by the work done by the active
force ($W_{F}$) in stretching the vesicle (Fig.3d). This energy is calculated
as the integrated change in length of the vesicle multiplied by the net active
force component that is aligned along the stretch direction (red arrows in
Fig.3a at $t=300$). As was shown for the two-arc configuration on a flat
substrate Sadhu2021 , the active work contributes a negative term to the total
effective energy of this configuration (Fig.3e), which is the sum
$W_{total}=W_{A}+W_{b}+W_{F}$, and acts to stabilize it. At a later stage
(time $\gtrsim 200$, Fig.3b), the adhesion energy is partially recovered as
the flat leading-edges are stretched along the axial directions by the active
forces, increasing their adhered area (Fig. 3j, 3h). Overall, the total
effective energy decreases (becomes more negative) over time (Fig. 3e), so the
process continues until full coiling.

Note that there is a low probability for the two arcs of the initially axial
vesicle (Fig. 3a, $t=0$) to reorient in the same direction along the
circumference. In such a case, the two arcs will merge to form a motile
crescent vesicle Sadhu_migration .

While the coiling process is therefore mainly driven by the active forces that
elongate the vesicle, performing active work, the above analysis does not
explain the origin of the initial reorientation of the leading-edges from the
axial to the circumferential alignment. In order to understand this stage, we
need to “zoom-in” on the dynamics of the leading-edges (during the process
shown in Fig.3a). We consider a section of the vesicle which contains the
leading-edge protein aggregate and the flat membrane protrusion that it forms
(Fig.3f). We define this leading-edge region as follows: we draw a plane
perpendicular to the direction of the net force of each protein arc, and place
it in a position such that all the proteins forming that arc are on one side
of this plane. This criteria is not sufficient when the shape of the arc is
highly coiled, so we use another constraint (Fig.3f at time$\gtrsim 300$): we
consider the leading-edge region for all nodes that are within a distance
$r_{max}$ from the center-of-mass (COM) of the proteins forming the arc, where
$r_{max}$ is the maximum distance of a protein in the arc from the COM of the
proteins. These criteria define a leading-edge membrane region that slightly
fluctuates in size over time (Fig. 3(g) inset, where we see that the biggest
change in the number of vertices occurs at time$\sim 300$).

At the beginning of the process the total force due to the proteins in each
arc is directed along the axial direction, and the angle of this force with
the cylinder’s axis increases until it is perpendicular ($\sim\pi/2$) at later
times (Fig.3g). The adhesion and bending energies per vertex of the membrane
within the leading-edge region are shown in Fig. 3(h,i). We plot the energies
per vertex to remove the effect of the variation in the number of vertices
over time (Fig.3g, inset). The adhesion energy fluctuates in the beginning but
there is an overall increase, while the bending energy slightly improves
throughout the process. The total value of adhesion and bending energies
without scaling by the number of vertices are shown in SI sec. S4, Fig. S6.

These changes arise from the flat membrane at the leading-edge region being
more stretched along the axis (the zero curvature direction) by the active
proteins when oriented circumferentially, thereby increasing the adhesion
energy and reducing the bending energy (Fig. 3j). In the initial axial
orientation the active forces of the proteins along the leading-edge are not
as effective in stretching the membrane sideways, as this involves strong
bending of the membrane around the fiber (Fig. 3j), and therefore encounters a
large bending energy penalty. From these observations we conclude that the
reorientation process of the leading-edge regions is mainly driven by locally
increasing the adhesion and decreasing the bending energies, while the global
coiling process of the whole vesicle is driven by the work done by the active
forces.

###  II.3 Theoretical predictions compared with experiments

We now use our theoretical model to make several predictions that we then test
in experiments.

In Fig.4a we plot the angular displacement ($\theta$) of the proteins on the
leading-edges of the membrane protrusions, where $\theta$ is defined as the
angle between the initial and subsequent location of a leading-edge protein
(measured from the center of the fiber) on the $X-Y$ plane. We next vary the
cross-section shape of the fiber by considering an elliptical cross section
having the same circumference as the circular fiber with a radius
$R=10l_{min}$. We find that above a critical aspect ratio (ratio of the semi-
major to the semi-minor axis of the ellipse $\sim 1.6$), the vesicle does not
coil but rather remains axially aligned (Fig. 4b, Movie-S14). The minimal
radius of curvature for this cross-section is $5~{}l_{min}$, which is a little
bit smaller than the radius at which coiling stops for a fiber with a circular
cross-section ($R\lesssim 7~{}l_{min}$ in Fig. 2). The coiling dynamics of the
leading-edges of the vesicle on a fiber with such a high aspect ratio is
inhibited, and the value of the angular position of the leading-edge
fluctuates around zero (Fig.4c), while on fibers with circular cross-section
the angular displacement of the leading-edges increases beyond $|2\pi|$, and
saturates when the vesicle fully coiled (Fig.4a).

We verify this prediction using our experimental results of the trajectories
of membrane ruffles for round and flat fibers (experimental images are shown
in Fig. 5; details of the experimental methods are in the Material and Method
section). Note that in the simulations we model the coiling of the leading-
edge that is adhered to the fiber surface, while in the experiments we could
visualize the motion of membrane ruffles that extend above the fiber surface.
Nevertheless, the motion of these ruffles appears to be correlated with and
reflects the qualitative features of the motion of the underlying leading-edge
membrane. We find that on a fiber with a round cross-section (diameter $200$
nm) the ruffles exhibit highly directed angular trajectories (Fig.4e), while
on the flat-ribbon fibers the trajectories fluctuate around zero (see
corresponding trajectories on the cell images in Fig. 5b), in close agreement
with the theoretical simulations (Fig.4a,c). The directed trajectories persist
for fibers of larger diameter ($1500$nm), but lose their directionality when
the fiber diameter is even larger ($3000$nm). These differences in the
experimental trajectories can also be quantified by their angular persistence
(Fig. 4f).

Figure 4: Membrane coiling dynamics on fibers of circular and elliptical
cross-sections. (a) Angular displacement ($\theta$) of the leading-edge of
vesicle with time for the circular fiber ($R=10l_{min}$). Different colors
represent different realizations. The inset shows the definition of angular
displacement $\theta$. Here, $P$ and $Q$ are the initial and final position of
a leading-edge protein on the $X-Y$ plane, and $\theta$ is the angular
displacement between them. We use here $E_{ad}=1.0k_{B}T$, $\rho=2.4\%$ and
$F=2.0k_{B}T/l_{min}$. The unit of time is $10^{5}$ MC steps. (b)
Configurations of vesicle with fibers of elliptical cross section. The coiling
ceases as the aspect ratio of the elliptical cross section ($R_{x}/R_{y}$)
increase. The circumference of the elliptical cross section is kept constant,
which is equal to the circumference of the circular cross-section ($=2\pi R$)
with $R=10.0l_{min}$. (c) Angular displacement of the leading-edge of the
vesicle as function of time, for an elliptical fiber of aspect ratio $1.6$.
Different colors represent different realizations. (d) Mean square angular
displacement (MSAD) with time for circular and elliptical fibers. The initial
growth of MSAD is $\sim t^{\alpha}$, where $\alpha=1$ represents diffusive
behavior while $\alpha=2$ shows ballistic nature. (e) Experimental data for
the angular position with time for different types of fibers. (f) Experimental
results for angular persistence for various types of circular and flat fibers.
(g) The value of power law exponent ($\alpha$) for the experimental data for
various fibers. (h) MSAD for the experimental case for various types of
fibers.

The difference between coiling dynamics on round and flat fibers can be
further quantified by plotting the mean square angular displacement (MSAD) for
both cases. The MSAD is expected to vary as $\sim t^{\alpha}$, where
$\alpha=1$ for a diffusive behavior, while $\alpha=2$ represents ballistic,
persistent coiling. In the simulations we find ballistic behavior for round
fibers (before the coiling saturates due to finite membrane area), while
diffusive for fibers with highly elliptic cross-section (Fig.4d). The
experimental data exhibits the same trends (Fig.4h), with ballistic dynamics
on the round fibers (of diameter $200$ nm and $1500$ nm), and diffusive motion
on the flat-ribbon fiber. On the round fibers with the largest diameter
($3000$nm), we observe mixed behavior. The values of the exponent $\alpha$,
extracted from the experimental data, are summarized in Fig.4g. The coiling
velocity for each of the experiments is shown in SI sec. S5, Fig. S7, compared
with the simulations.

The experimental images of the cells’ leading-edges and the trajectories of
the ruffles are shown in Fig. 5 (also see Movie-S15-S19). Fig. 5A shows the
isometric view of the cells leading-edge. The protrusions are highly dynamic
for the round fiber of $D=200~{}nm$, while less for the flat-ribbon fiber
(Fig. 5B). The kymographs in Fig. 5C clearly demonstrate that the ruffles on
the round fibers coil around the fiber and move from side to side of the fiber
axis, while on the flat-ribbon fibers they stay only on one side and do not
wrap around the axis.

Figure 5: Coiling at leading-edges for cells spread on fibers with different
diameters. The round polystyrene fibers have diameter of $200$nm, $1500$nm and
$3000$nm, while the “Flat Ribbon” fibers are produced by flattening a $200nm$
fiber, increasing its cross-sectional width to approximately
$\pi~{}D~{}\approx~{}600~{}nm$ with a thickness $t~{}\approx~{}50~{}nm$. (A)
Maximum intensity projections of the inclined volumetric scan of the cell with
Dual-View Light Sheet microscopy. (B) Typical coiling trajectories of membrane
ruffles, extracted by cross-sectional manual tracking. The color lines denote
the displacements of the tracked ruffle, with the trajectories ending with the
dot. (C) Kymographs for the ruffles shown in B. These are illustrating the
movement of the membrane boundary along a line placed at the leading-edge of
the cell as shown in the dashed line of the illustration on the left.

##  III Discussion

Cells often encounter extra-cellular fibers on which they adhere, spread and
migrate. We show that the tendency of the leading-edge of such cells to coil
around adhesive fibers can be understood using a model with a minimal set of
components: membrane with curved membrane proteins that induce outwards active
forces (representing cytoskeletal activity), and adhesion. Within this model
the coiling process emerges spontaneously, and is driven by the physics of
minimizing the free energy and the active work. This physics-based model
predicts that the coiling will be inhibited when the radius of curvature of
the fiber is too small, which prevents coiling around flat ribbons due to high
membrane bending energy. This prediction is verified in our experiments.
Furthermore, the essential role of the curved membrane proteins in our model
can explain the reduced coiling observed in cells with reduced amount of such
curved and actin-related proteins Nain-unpublished .

There are obvious biological consequences for the coiling process. Cells may
form better overall grip when coiling on fibrillar structures (i.e., ECM and
cellular processes), aiding cell motility and intercellular interactions. For
example, cancer cells exhibit enhanced coiling activity on fibers
mukherjee2019cancer ; nain2022 . Furthermore, cells migration on tube-like
cellular structures during tissue development and organogenesis, as well as
during cancer progression, may utilize membrane coiling. One such example is
the process of myelination, during which glial cells of the vertebrate central
and peripheral nervous systems produce a multi lamellar substance called
myelin around axons, thereby allowing fast nerve conduction Nave2014 ;
Wilson2021 . In the peripheral nervous system, Schwann cells myelination is a
process that requires these cells to efficiently coil around fiber-like axons
Gatto2003 ; Pedraza2009 . In order to closely visualize coiling of Schwann
cells on axons, we used a transgenic mouse which expressed a GFP-tagged
myelin-specific membrane protein ERB2006  to generate Schwann cell-neuron
myelinating cultures with neurons that expressed TdTomato (red fluorescence)
in their cytoplasm. Time lapse imaging of these cultures clearly showed that
already at the initial Schwann cell-axon interaction (Fig.6A, Movie-S20) the
Schwann cell sent thin processes that coiled around the axons that were to be
myelinated. During the myelination process itself (Fig.6B, Movie-S21), we
visualized slower and more prominent spiraling of the Schwann cell membrane
around the axon, as demonstrated by a kymograph (Fig. 6C), which probably
represents wrapping of the inner myelin layer, as known to occur during myelin
formation Snaidero2014 . It is thus clear that coiling of Schwann cell
membranes around axons is a main feature of the unique inter-cellular
interactions between Schwann cells and the cylindrical axonal processes.

Our theoretical work indicates that there is a fundamental physical mechanism
that leads to membrane coiling dynamics, which can then be further tuned and
regulated by biological molecular signaling. Furthermore, myelination is known
to occur on axons for which diameter exceeds a critical threshold Nave2014 .
Our results reveal a physical mechanism that inhibits coiling, and may
correspondingly prevent myelination around very thin fibers in the vertebrate
nervous system.

In conclusion, we propose a physics-based mechanism for the tendency of
adhering cellular membranes to coil around fibers. The mechanism is driven by
the out-of-equilibrium (active) nature of the forces exerted by the
cytoskeleton on the membrane, and the feedback between curved membrane
proteins, adhesion and membrane bending. This mechanism may underlie crucial
processes in biology, and remains to be tested by more detailed molecular
studies.

Figure 6: Coiling events involved in inter-cellular interactions during
vertebrate myelin development. Time lapse imaging of mouse DRG myelinating
culture showing (A) initial contact between a thin process of pre-myelinating
Schwann cell (green) spiraling around an axon (red). Time intervals between
images are $30$ minutes. (B) As myelination proceeds, a thicker Schwann cell
process wraps around an axon at a slower speed. Time intervals between images:
$1$ hour. Arrows point at clear spiral turns, as they rotate. (C) Kymograph
showing fluorescence intensity along a fixed line across the spiral shown in
(B) (depicted as a dashed white line) over the duration of $4$ hours in 15
minute intervals. Scale bars: $10~{}\mu m$. The kymograph clearly shows the
apparent upwards motion of the fluorescent Schwann cell process, as the spiral
shape rotates around the axon.

##  IV Materials and methods

###  IV.1 Theoretical model

We consider a three-dimensional vesicle, which is described by a closed
surface having $N$ vertices, each of them is connected to its neighbors with
bonds, and forms a dynamically triangulated, self-avoiding network, with the
topology of a sphere miha2019 ; Sadhu2021 ; Sadhu_phagocytosis . The vesicle
is placed on a fiber, with which the vesicle has a uniform attractive contact-
interaction (adhesion), as shown in Fig. 7. The vesicle energy has the four
following contributions: The bending energy is given by,

$W_{b}=\frac{\kappa}{2}\int_{A}(C_{1}+C_{2}-C_{0})^{2}dA,$ (1)

where $\kappa$ is the bending rigidity, $C_{1}$, $C_{2}$ are the principal
curvatures and $C_{0}$ is the spontaneous curvature. We consider the
spontaneous curvature $C_{0}$ to be non-zero at the location of the curved
proteins, while in the absence of any curved proteins, the spontaneous
curvature is considered to be zero. We consider only convex curved proteins in
our simulation, such that their spontaneous curvature $C_{0}=1.0l_{min}^{-1}$.
The protein-protein interaction energy is given by,

$W_{d}=-w\sum_{i<j}{\cal H}(r_{0}-r_{ij}),$ (2)

where, ${\cal H}$ is the Heaviside step function,
$r_{ij}=|\overrightarrow{r}_{j}-\overrightarrow{r}_{i}|$ is the distance
between proteins and $r_{0}$ is the range of attraction, $w$ is the strength
of attraction. The range of attraction is chosen in a way so that only the
proteins that are in the neighboring vertices can interact with each other.
The active energy is given by,

$W_{F}=-F\sum_{i}\hat{n_{i}}.\overrightarrow{r}_{i},$ (3)

where, $F$ is the magnitude of the force, $\hat{n_{i}}$ is the outward normal
vector of the vertex that contains a protein and $\overrightarrow{r}_{i}$ is
the position vector of the protein. These forces are ”active” since they give
an effective energy (work) term that is unbounded from below and thereby drive
the system out-of-equilibrium. If the proteins are distributed
inhomogeneously, there will be a net force in a particular direction. However,
since we only simulate vesicles that are adhered to the rigid fibers, and the
fiber is fixed in its location, this links our vesicle to the lab frame
thereby restoring momentum conservation (fixing the fiber acts as an infinite
momentum reservoir).

Finally, the adhesion energy, which is due to the attractive interaction
between the fiber and the vesicle, is given by,

$W_{A}=-\sum_{i^{\prime}}E_{ad},$ (4)

where $E_{ad}$ is the adhesion strength, and the sum runs over all the
vertices that are adhered to the fiber samo2015 ; miha2019 ; Sadhu2021 . By
‘adhered vertices’, we mean all such vertices, whose perpendicular distance
from the surface of the fiber are less than $l_{min}$. Thus, the total energy
of the system is given by,

$W=W_{b}+W_{d}+W_{F}+W_{A}$ (5)

We update the vesicle with mainly two moves, (1) Vertex movement and (2) Bond
flip. In a vertex movement, a vertex is randomly chosen and attempt to move by
a random length and direction, with the maximum possible distance of
$0.15l_{min}$. In a bond flip move, a single bond is chosen, which is a common
side of two neighboring triangles, and this bond is cut and reestablished
between the other two unconnected vertices samo2015 ; miha2019 ; Sadhu2021 .
The maximum bond length is chosen to be $l_{max}=1.7l_{min}$. The fiber is
assumed to be infinite in length in the axial direction ($Z$-axis), while
having a finite cross-section (either circular or elliptical) on the $X-Y$
plane (Fig. 7).

We update the system using Metropolis algorithm, where any movement that
increases the energy of the system by an amount $\Delta W$ occurs with rate
$exp(-\Delta W/k_{B}T)$, otherwise it occurs with rate unity.

We used in this study a vesicle of total number of vertices, $N=3127$ (radius
$\sim 20~{}l_{min}$), where $l_{min}$ is the unit of length in our model, and
defines the minimum bond length. The bending rigidity $\kappa=20~{}k_{B}T$,
the protein-protein attraction strength $w=1~{}k_{B}T$, and $\rho=N_{c}/N$ is
the protein density, where $N_{c}$ are the number of vertices occupied by
curved membrane proteins with spontaneous curvature:
$C_{0}=1.0~{}l_{min}^{-1}$. We do not conserve the vesicle volume here, while
the membrane area is well conserved ($\Delta A/A<1\%$) miha2019 .

Note that our simulation length scale $l_{min}$ does not have any
correspondence with a real length. By assigning a different real length to the
basic length-scale of the simulation ($l_{min}$) we do not in fact change the
dynamics. The effect of such a change in length scale will be just to rescale
the parameters ($E_{ad}$, $F$, $C_{0}$ etc.) according to the choice of
$l_{min}$. The absolute scaling of $l_{min}$ does not therefore make any
qualitative change in our simulations, since all the energies are scale-
invariant with our definition of $E_{ad}$ and $F$, while the bending energy is
always scale-invariant (intensive energy).

Parameters used in simulation
---
Symbol | Definition | Unit
$N$ | Number of vertices | NA
$R$ | Radius of the round fiber | $l_{min}$
$R_{x}~{}(R_{y})$ | Semi-major (minor) axis of the flat fiber | $l_{min}$
$E_{ad}$ | Adhesion strength |  $k_{B}T$ (per node)
$\rho$ | Density of curved proteins | NA
$w$ | Strength of protein-protein binding | $k_{B}T$
$F$ | Active protrusive force | $k_{B}T/l_{min}$
$C_{0}$ | Spontaneous curvature | $l_{min}^{-1}$
Table 1: List of parameters used in our simulation Figure 7: Schematic
representation of our theoretical model. We consider a three dimensional
vesicle, and an adhesive fiber of circular as well as elliptical cross-
section. The red dots on the vesicle represent the curved proteins, while the
blue regions are the bare membrane.

###  IV.2 Imaging of the coiling of leading-edge protrusions on suspended
fibers

####  IV.2.1 Scaffold Preparation

Using the previously reported non-electrospinning STEP technique Nain2009 ;
Nain2013 ; nain2014 , suspended fiber nanonets composed of 200 or 500 nm
diameter fibers spaced 20 μm apart were deposited orthogonally on 2 μm
diameter fibers spaced 300 μm apart. Nanofibers were manufactured from
solutions of polystyrene (MW: 2,500,000 g/mol; Category No. 1025; Scientific
Polymer Products, Ontario, NY, USA) dissolved in xylene (X5-500; Thermo Fisher
Scientific, Waltham, MA, USA) in 6 and 9 wt% solutions. 1 and 3.5 μm diameter
fibers were manufactured from 2 and 5 wt% of high molecular weight polystyrene
(MW: 15,000,000 g/mol, Agilent Technologies, Santa Clara, CA, USA) and equally
spaced at approximately 60 µm. The polymeric solutions were extruded through
micropipettes with an inside diameter of 100μm (Jensen Global, Santa Barbara,
CA, USA) for deposition of fibers on a hollow substrate. All fiber networks
were crosslinked at intersection points using a custom fusing chamber, to
create fixed-fixed boundary conditions. 200nm diameter fibers were made into
flat ribbons (of width $~{}\pi D$, where $D$ is the diameter of the original
fiber of circular cross-section) as described in koons2017 .

####  IV.2.2 Cell Culturing

Mouse Muscle Myoblasts (C2C12s) expressing GFP actin were a gift from the
Konstantopoulos Lab at Johns Hopkins University. Cells were cultured on Petri
dishes using Dulbecco’s Modified Eagle Medium (DMEM, Gibco, Thermo Fisher
Scientific) with 10% Fetal Bovine Serum (FBS, Invitrogen, Carlsbad, CA, USA)
and 1% Penicillin-Streptomycin. STEP-spun scaffolds were placed on 100x20mm
Petri dishes and disinfected with 70% ethanol, then coated with 6µg/ml of
Rhodamine Fibronectin (Cat. # FNR01, CYTOSKELETON, Denver, CO, USA) by
incubating at $37^{o}C$ for 1hr. The trypsinized and resuspended cells were
seeded onto STEP-spun scaffolds, allowed to attach for at least two hours and
the wells were flooded with 2mL of DMEM +10% FBS.

####  IV.2.3 Cell Imaging with Dual-View Light Sheet Microscopy

The seeded samples with DMEM were PBS washed, then flooded with Live-Cell
Imaging Solution (Invitrogen, Carlsbad, CA, USA) for imaging using Dual-View
Plane Illumination Microscopy (diSPIM) Kumar2014 ; Wu2013 . Samples were
mounted in the diSPIM and kept at $37^{o}C$ after calibrating the light-sheet
movement to piezo step factor using a micromanager plugin Ardiel2017 . Cells
were volumetrically imaged (two views), up to 120 time points at 1.5, 2.5, and
3.5-second intervals constrained by the cell size, the number of slices, and
location in the suspended nanonets.

####  IV.2.4 Cell Imaging with Dual-View Light Sheet Microscopy

Z stacks acquired from the diSPIM were background subtracted, bleach
corrected, cropped, and sorted in Fiji Schindelin2012 . The orthogonal views
(SPIM A and SPIM B) were fused using a GPU optimized pipeline as described in
Guo 2020 Guo2020 . Fused stacks with isotropic resolution were rotated with
the TransformJ plugin for visualization and analysis. Maximum intensity
projections derived from the volumetric time series were used to manually
track the protrusive activity at the coiling fronts of cells. RStudio was used
to generate all plots and statistical comparisons of means using Kruskal-
Wallis test.

###  IV.3 Imaging of early stage myelination process

For the generation of Schwann cell-dorsal root ganglia (DRG) neuron
myelinating cultures we used mice expressing S-MAG-GFP (a transgene expressed
specifically in myelinating cells membranes ERB2006 ). DRG cultures were
prepared from mouse embryos at day $13.5$ of gestation. DRGs were dissociated
and plated at a density of $4\times 10^{4}$ per chamber (Lab-Tek coverglass
system Nunc #155411), coated with Matrigel (Becton Dickinson) and Poly-L-
lysine. Cultures were grown for $2$ days in Neurobasal medium supplemented
with B-27, glutamax, penicillin /streptomycin and $50~{}ng/ml$ NGF. At day $2$
post seeding cultures were infected with a lentivirus carrying a cytoplasmic
Tdtomato reporter, resulting in neuronal expression of TdTomato. Cultures were
then grown for 4 additional days in BN medium containing Basal medium-Eagle,
ITS supplement, glutamax, $0.2\%$ BSA, $4~{}mg/ml$ D-glucose, $50~{}ng/ml$ NGF
and antibiotics. To induce myelination, cultures were grown in BNC, namely a
BN medium supplemented with $15\%$ heat inactivated fetal calf serum
(replacing the BSA) and $50~{}\mu g/ml$ L-ascorbic acid.

Cultures were imaged at $7$ days with myelinating medium (for fluorescence
imaging, at a frequency of $4$ frames per hour for $65$ or $19$ hours).
Fluorescence images were obtained using a confocal microscope (LSM700 confocal
microscope Carl Zeiss) $488$-nm and $555$-nm laser lines. Confocal time-lapse
images were captured using a Plan-Apochromat $20\times/0.8$ M27 objective
(Carl Zeiss), at temperature , CO2 and humidity controlled conditions. Image
capture was performed using acquisition Blocks (Carl Zeiss Zen 2012). Images
and movies were generated and analyzed using Zeiss Zen 2012 and Adobe
Photoshop CC 2019.

##  V Acknowledgment

ASN acknowledges partial funding support from National Science Foundation
(NSF, Grant No. 1762468). ASN and BB acknowledge the Institute of Critical
Technologies and Science (ICTAS) and Macromolecules Innovative Institute (MII)
at Virginia Tech for their support in conducting this study. HS acknowledges
the support from the intramural research program of the National Institute of
Biomedical Imaging and Bioengineering within the National Institutes of
Health. The basic version of the code (for theoretical modelling) is provided
by Samo Penič, which is further edited by RKS for the implementation of the
present model. AI acknowledges the support from the Slovenian Research Agency
(ARRS) through Programme No. P2-0232 and project No. No. J3-3066.

##  VI Supplementary movies

All the supplementary movies are available online in the link:
https://app.box.com/s/slkec3eu8h34fo6p9s414s0km4a8jgw5

##  S-0 Supplementary informations

###  S-0.1 Results for passive vesicle

We present here the results for the adhesion and spreading of protein-free
vesicles on cylindrical substrates, for different fiber radius and strength of
adhesion (Fig. S-1). As expected, the adhered area fraction increases strongly
with both fiber radius and adhesion energy, with the transition to complete
adhesion determined by the balance between the bending and adhesion energies.
At large adhered area fractions, the bending energy per vesicle-substrate area
balances the adhesion area, with the complete adhesion regime occurring when:
$\kappa/R^{2}\leq E_{ad}$, shown by the yellow dashed line in Fig. S-1.

Figure S-1: Phase diagram $E_{ad}-R$ plane for protein-free case ($\rho=0$).
Background color is showing the adhered fraction of vesicle. Adhered fraction
increases either with $R$ or $E_{ad}$.

We next show our results with passive curved proteins ($F=0$) and a given
radius of fiber $R=10~{}l_{min}$ in Fig. S-2. With the introduction of curved
passive proteins, the complete adhesion regime is extended to lower adhesion
energies, as was observed for flat substrates Sadhu2021 . For a given small
$E_{ad}$, as we increase protein density $\rho$, the vesicle gets adhered more
and more. For larger $E_{ad}$, however, the strong adhesion is achieved even
without (or small) proteins. The yellow horizontal dashed line is showing the
transition given by $\kappa/R^{2}\leq E_{ad}$ (for a given $R$) above which
the adhesion energy dominates.

Figure S-2: Phase diagram $E_{ad}-\rho$ plane for vesicle with passive
proteins ($F=0$). Background color is showing the adhered area fraction. The
adhered fraction increases with $E_{ad}$ and $\rho$.

###  S-2.2 Analytical estimation of transition lines

Here, we present our simple analytical estimation of all the transition lines
separating different phases (shown in Fig. 2d, main text).

####  S-2.2.1 Transition to a pancake shape (phase-I to II or phase-IV to III)

Let us consider the surface area of the vesicle is $A$. If we make this a flat
pancake-like, the radius of this circular face will be given by the relation,

$\pi R_{p}^{2}\simeq A/2$ (S-2)

or,

$R_{p}\simeq\sqrt{A/2\pi}$ (S-2)

The perimeter of the circular face of the pancake-like shape is,

$2\pi R\simeq\sqrt{2\pi A}$ (S-2)

If the average size of a protein is, $l_{av}=(l_{min}+l_{max})/2$, then, the
number of proteins that will fit in that perimeter is given by,

$N_{p}\simeq\frac{\sqrt{2\pi A}}{l_{av}}$ (S-2)

Now, the average surface area of the vesicle is,

$A=2N\frac{\sqrt{3}}{4}l_{av}^{2}$ (S-2)

where, $N$ is the number of nodes, and $2N$ is the approximate number of
triangles. Since, $N=3127$, and $l_{av}=1.35~{}l_{min}$, the total number of
proteins that will fit into the rim of circular pancake is,

$N_{p}=\frac{\sqrt{2\pi A}}{l_{av}}\simeq 130$ (S-2)

The density of proteins is thus,

$\rho_{p}(\%)=(100\times 130)/N\simeq 4\%$ (S-2)

####  S-2.2.2 Transition from “hanging” pancake (phase-II) to “wrapped”
pancake (phase-III)

Let us assume the vesicle forms a circular pancake-like shape with the radius
of $R_{p}$, such that the total area of the vesicle, $A=2\pi R_{p}^{2}$. Now,
consider the case where the pancake just touching the cylinder and remains
unadhered. The adhesion energy in this case is,

$W_{A}=-E_{ad}\frac{2dR_{p}}{l^{2}_{av}}$ (S-2)

Here, $l_{av}$ is the average length of the bond,
$l_{av}=(l_{min}+l_{max})/2$. Here, we assume that the width that is in
contact with the cylinder is of the length $d=l_{min}$, such that the number
of adhered nodes are $2R_{p}/l^{2}_{av}$. After the pancake is fully adhered,
the new adhesion energy and the bending energy are,

$W_{A}=-E_{ad}\frac{\pi R_{p}^{2}}{l^{2}_{av}}$ (S-2)

and the extra bending energy cost for this transition,

$W_{b}=\kappa\pi R_{p}^{2}\frac{1}{R^{2}}$ (S-2)

The condition for the transition is,

$\frac{E_{ad}}{l^{2}_{av}}(\pi R_{p}-2)=\kappa\frac{\pi R_{p}}{R^{2}}$ (S-2)

which gives,

$R^{2}=\frac{\kappa}{E_{ad}}\frac{l^{2}_{av}}{\left(1-\frac{2}{\pi
R_{p}}\right)}$ (S-2)

Assuming $R_{p}\gg 1$, we have,

$R\sim\sqrt{\kappa/E_{ad}}l_{av}\sim 6l_{min}$ (S-2)

####  S-2.2.3 Coiling transition (phase-I to phase-IV)

For small $\rho$, the vesicle either forms elongated shapes axially aligned,
or coiled shape. For small $R$, the vesicle aligned along the axial direction
and for large $R$, it forms a circumferential (coiled) shape. Let us assume a
spherical vesicle having zero force (Fig. S-3a(i)). As the force increases,
the vesicle will get stretched (Fig. S-3a(ii)). Using the energy minimization,
and area conservation, we can get the elongated shape, assuming it to be
cylindrical. The area conservation states,

$4\pi R_{0}^{2}\simeq 2\pi r_{c}h_{t}$ (S-2)

where, we neglect the area for the two circular edges. Now, assume that for
spherical vesicle, only a single vertex of adhered, while for cylindrical
shape, a linear chain of vertices are adhered along the axial direction. Thus,
from energy minimization,

$(N_{c}R_{0}F+E_{ad}-8\pi\kappa)r_{C}^{2}-2R_{0}^{2}(E_{ad}/l_{av}+N_{c}F/2)-8\pi\kappa
R_{0}^{2}(N_{c}R_{0}F+E_{ad}-8\pi\kappa)=0$ (S-2)

The solution of this equation will give two values of $r_{c}$ among which, one
is greater than $R_{0}$ and hence is unphysical. We consider the other
solution We can also calculate $h_{t}$ from area conservation.

Now, let us assume that for large $R$, when the vesicle goes to a coiled
shape, the value of $r_{c}$ remains same, and the shape becomes a combination
of cylinder and two flat circular shapes (Fig. S-3a(iii)). The radius of the
flat circular shapes will be determined by the protein density,

$2\pi r-2r_{c}=N_{c}l_{av}/2$ (S-2)

or,

$r=(N_{c}l_{av}/2+2r_{c})/2\pi$ (S-2)

From area conservation,

$4\pi R_{0}^{2}=2\pi hr_{c}+4\pi r^{2}$ (S-2)

and from energy minimization,

$-E_{ad}h_{t}/l_{av}+\frac{\kappa}{2}2\pi
r_{c}h_{t}\frac{1}{r_{c}^{2}}=-E_{ad}(h/l_{av}+2\pi
r^{2}/l_{av}^{2})+\frac{\kappa}{2}2\pi
r_{c}h\frac{1}{r_{c}^{2}}+\frac{\kappa}{2}4\pi R_{0}^{2}\frac{1}{R^{2}}$ (S-2)

$R^{2}=\frac{\kappa A}{2}[E_{ad}(h/l_{av}+2\pi
r^{2}/l_{av}^{2}-h_{t}/l_{av})+\pi\kappa/r_{c}(h_{t}-h)]^{-1}$ (S-2)

where, $R$ is the radius of the cylinder. We solve the above equation
numerically in the $R-\rho$ plane, and show in Fig. S-3(b) with green solid
line, and compare with simulation results with dashed line-points.

Figure S-3: Analytical estimation of transition line for coiling transition
(phase-I to phase-IV). (a) Schematic representation of steps of analytical
calculations. (i) For the vesicle without any force, the shape is spherical.
(ii) For small $R$, when force is applied, the vesicle shape changes to
cylindrical one. (iii) For larger $R$, the vesicle shape changes to coiling
shape, having a combination of cylindrical and circular shape. The red lines
are representing the curved proteins. (b) Comparison of analytical estimation
with the simulation. Other simulation parameters are same as in Fig. 2, main
text.

###  S-3.3 Quantification of axial and circumferential alignment

In our main text, we quantify the axial and circumferential alignment of the
vesicle by measuring the variance in the distribution of vertices along the
axial direction and the distribution of angle along the circumferential
direction respectively. Here, we show the full distribution of the vertices
along the axial and circumferential direction in Fig. S-4 and Fig. S-5
respectively.

Figure S-4: Distribution of vertices along the axial direction. (a)
$\rho=3.2\%$ and $R=3.0l_{min}$. (b) $\rho=3.2\%$ and $R=8.0l_{min}$. (a)
$\rho=9.6\%$ and $R=3.0l_{min}$. (a) $\rho=9.6\%$ and $R=8.0l_{min}$.  Figure
S-5: Angular distribution of the vertices along the circumferential direction.
(a) $\rho=3.2\%$ and $R=3.0l_{min}$. (b) $\rho=3.2\%$ and $R=8.0l_{min}$. (a)
$\rho=9.6\%$ and $R=3.0l_{min}$. (a) $\rho=9.6\%$ and $R=8.0l_{min}$.

###  S-5.4 Reorientation process: Energy with time without scaling per vertex

Here, we show the adhesion and bending energies for an arc without scaling by
the total number of vertices that are forming the arc, in Fig. S-6(a-b). We
also show the energy of the rest of the cylindrical part of the vesicle (after
subtracting the contribution due to the other two arcs) in Fig. S-6(c-d). We
note that the adhesion energy is increasing throughout and bending energy is
also larger in the final configuration.

Figure S-6: (a) Adhesion energy with time for an arc. (b) Bending energy with
time for an arc. (c) Adhesion energy with time for the rest cylindrical part.
(d) Bending energy with time for the rest cylindrical part. Other simulation
parameters are same as in Fig. 3, main text.

###  S-6.5 Coiling speed for different fiber radius

In Fig. S-7 we compare the coiling velocity of the leading edge protrusion
measured from experiments with the simulation data. The simulation data
indicates that over the fiber radii that we can test, the coiling speed does
not change significantly (slopes of the graphs in Fig. S-7b). However, in the
experiments the coiling speed was found to decrease for increasing radii, most
significantly for small radius (Fig. S-7d).

Figure S-7: Coiling speed for different fiber radius in simulations, and
compared with experiments. (a) The overall angular-length-span of the vesicle
coil as function of time. (b) The displacements of the leading-edge arcs along
the angular direction, as function of time. (c) Radial length of coil
trajectory measured from experiments. (d) Coiling velocity from the
experimental data, for each of the four cases.

##  S-7 Supplementary movies

All the supplementary movies are available online in the link:
https://app.box.com/s/slkec3eu8h34fo6p9s414s0km4a8jgw5

  * •

Movie-S1Experimental movie corresponding to Fig. 1A. Deconvolved isometric
render in ImageJ 3D Viewer of a C2C12 expressing GFP actin spread on a
rhodamine fibronectin coated $200~{}nm$ fiber. Scale bar shown is for $10\mu
m$.

  * •

Movie-S2Experimental movie corresponding to Fig. 1B. Zoomed-in perspective of
leading/trailing edge of the cell as it migrates and coils around the 200nm
fiber. Scale bar shown is for $5\mu m$

  * •

Movie-S3Experimental movie corresponding to Fig. 1C. Zoomed-in perspective of
leading/trailing edge of the cell as it migrates and coils around the 200nm
fiber. Scale bar shown is for $5\mu m$.

  * •

Movie-S4Spreading of protein-free vesicle on cylindrical fiber of smaller
radius. The parameter values are : $E_{ad}=1.5~{}k_{B}T$, $R=5.0~{}l_{min}$.

  * •

Movie-S5Spreading of protein-free vesicle on cylindrical fiber of larger
radius. The parameter values are : $E_{ad}=1.5~{}k_{B}T$, $R=15.0~{}l_{min}$.

  * •

Movie-S6Spreading of a vesicle with passive proteins (low density) on
cylindrical fiber. The parameters are: $E_{ad}=1.0~{}k_{B}T$, $\rho=1.6\%$,
$R=10.0~{}l_{min}$.

  * •

Movie-S7Spreading of a vesicle with passive proteins (medium density) on
cylindrical substrate. The parameters are: $E_{ad}=1.0~{}k_{B}T$,
$\rho=3.2\%$, $R=10.0~{}l_{min}$.

  * •

Movie-S8Spreading of a vesicle with passive proteins (high density) on
cylindrical substrate. The parameters are: $E_{ad}=1.0~{}k_{B}T$,
$\rho=6.4\%$, $R=10.0~{}l_{min}$.

  * •

Movie-S9Spreading of a vesicle with active proteins with small $R$ and small
$\rho$ (Phase-I). The parameters are: $E_{ad}=1.0~{}k_{B}T$, $\rho=3.2\%$,
$R=4.0~{}l_{min}$, and $F=2.0~{}k_{B}T/l_{min}$.

  * •

Movie-S10Spreading of a vesicle with active proteins with small $R$ and large
$\rho$ (Phase-II). The parameters are: $E_{ad}=1.0~{}k_{B}T$, $\rho=6.4\%$,
$R=4.0~{}l_{min}$, and $F=2.0~{}k_{B}T/l_{min}$.

  * •

Movie-S11 Spreading of a vesicle with active proteins with large $R$ and large
$\rho$ (Phase-III). The parameters are: $E_{ad}=1.0~{}k_{B}T$, $\rho=6.4\%$,
$R=10.0~{}l_{min}$, and $F=2.0~{}k_{B}T/l_{min}$.

  * •

Movie-S12Spreading of a vesicle with active proteins with large $R$ and small
$\rho$ (Phase-IV). The parameters are: $E_{ad}=2.0~{}k_{B}T$, $\rho=2.4\%$,
$R=10.0~{}l_{min}$, and $F=2.0~{}k_{B}T/l_{min}$.

  * •

Movie-S13Axial to circumferential (coiling) transition. The parameters are:
$E_{ad}=1.0~{}k_{B}T$, $\rho=2.4\%$, $R=10.0~{}l_{min}$, and
$F=2.0~{}k_{B}T/l_{min}$.

  * •

Movie-S14Spreading of vesicle with active proteins on a cylinder with
elliptical cross-section. The parameters are: $E_{ad}=1.0~{}k_{B}T$,
$\rho=3.2\%$, $R_{x}=12.0~{}l_{min}$, $R_{y}=7.77~{}l_{min}$, and
$F=2.0~{}k_{B}T/l_{min}$.

  * •

Movie-S15Experimental movie corresponding to Fig. 5a, 200nm, leading-edge 1.
Maximum intensity projection of the leading-edge volume used for rotational
analysis. Dots and lines are overlayed paths using ImageJ manual tracking
plugin.

  * •

Movie-S16Experimental movie corresponding to Fig. 5a, 200nm, leading-edge 2.
Maximum intensity projection of the leading-edge volume used for rotational
analysis. Dots and lines are overlayed paths using ImageJ manual tracking
plugin.

  * •

Movie-S17Experimental movie corresponding to Fig. 5b, flat ribbon. Maximum
intensity projection of the leading-edge volume used for rotational analysis.
Dots and lines are overlayed paths using ImageJ manual tracking plugin.

  * •

Movie-S18Experimental movie corresponding to Fig. 5c, $1500~{}nm$. Maximum
intensity projection of the leading-edge volume used for rotational analysis.
Dots and lines are overlayed paths using ImageJ manual tracking plugin.

  * •

Movie-S19Experimental movie corresponding to Fig. 5d, $3000~{}nm$. Maximum
intensity projection of the leading-edge volume used for rotational analysis.
Dots and lines are overlayed paths using ImageJ manual tracking plugin.

  * •

Movie-S20Time lapse imaging of mouse DRG myelinating culture showing initial
contact (arrow) between a pre-myelinating Schwann cell (green) and an axon
(red).Images were taken at time intervals of 15min. The Schwann cell process
spirals around the axon at a speed of aprx $90$ minutes per round . Figure 6A
depicts frames #68,70,72,74 of this movie.

  * •

Movie-S21Time lapse imaging of mouse DRG myelinating culture, taken time
intervals of $15$ minutes, showing a myelinating Schwann cell membrane (green,
arrow) wrapping around an axon (red) at a speed of aprx $180$ minutes per
round, generating a prominent dynamic spiral. Figure 6B depicts frames
52,56,60,64 of this movie.

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