# Mechanistic insights into Z-ring formation and stability: A Langevin
dynamics approach to FtsZ self-assembly

Rajneesh Kumar<EMAIL_ADDRESS>Institute of Physics, Sachivalaya
Marg, Sainik School, Bhubaneswar 751005, India  Ramanujam Srinivasan
<EMAIL_ADDRESS>National Institute of Science Education and Research,
Khurda 752050, Odisha, India  Homi Bhabha National Institutes (HBNI), Training
School Complex, Anushakti Nagar, Mumbai, India 400094  Debasish Chaudhuri
<EMAIL_ADDRESS>Institute of Physics, Sachivalaya Marg, Sainik School,
Bhubaneswar 751005, India  Homi Bhabha National Institutes (HBNI), Training
School Complex, Anushakti Nagar, Mumbai, India 400094

###### Abstract

The tubulin-like protein FtsZ is crucial for cytokinesis in bacteria and many
archaea, forming a ring-shaped structure called the Z-ring at the site of cell
division. Despite extensive research, the formation of Z-rings is not entirely
understood. We propose a theoretical model based on FtsZ’s known filament
structures, treating them as semiflexible polymers with specific mechanical
properties and lateral inter-segment attraction that can stabilize ring
formations. Our Langevin dynamics simulations reveal various morphological
phases, including open helices, chains, rings, and globules, capturing
experimental observations in the fission yeast model using FtsZ from different
bacterial species or mutants of Escherichia coli. Using the theoretical model,
we explore how treadmilling activity affects Z-ring stability and identify a
spooling mechanism of ring formation. The active ring produces contractile,
shear, and rotational stresses, which intensify before the Z-ring transitions
to an open helix at high activity.

##  I Introduction

FtsZ is one among the diverse family of bacterial cytoskeletal proteins that
exhibits distinct homologies to eukaryotic tubulin [1, 2, 3, 4]. In
prokaryotic cells, e.g., bacteria and many archaea, FtsZ (filamenting
temperature-sensitive mutant Z) proteins assemble into a ring, the Z-ring, at
the future division site, and it attaches to the cell membrane [5, 6]. This
ring provides the backbone for the formation of divisome [7, 8], a complex
multi-protein structure that recruits other proteins required for septum
formation between dividing cells. Moreover, the Z-ring itself may generate a
constriction force helping cytokinesis [9].

Assembly of FtsZ into a ring-like structure requires polymerization of the
monomeric units of FtsZ, which, like tubulin, depends on the binding of the
nucleotide GTP [10, 11, 12, 13, 14]. Although GTP hydrolysis triggers the
disassembly of FtsZ filaments, the assembly and polymerization of FtsZ itself
is required for the GTPase activity. This is because GTP is sandwiched in the
interface between two subunits of FtsZ, and while one subunit binds GTP, the
catalytic residue necessary to hydrolyze the terminal phosphate in GTP is
provided by the adjoining subunit [13, 15, 16]. GTP hydrolysis rates in FtsZ
polymers regulate the turnover rates of the filaments [17, 18, 19] and, more
recently, have been shown to drive treadmilling activity in FtsZ filaments
[20, 21, 22]. In vivo, FtsZ filaments are dynamic. They are seen to organize
as spiral structures along the axis of the cell during the cell elongation and
condense into a ring (Z-ring) during cytokinesis, which is tightly coupled to
the cell cycle progression [23, 24, 25]. In Bacillus subtilis, such spiral
FtsZ filaments have been seen to coalesce into a ring during sporulation [23].
Other studies have also suggested a spiral or helical intermediate in the
formation of the cytokinetic FtsZ ring [26]. However, despite numerous earlier
studies, the exact mechanisms of FtsZ ring formation are not fully understood.

FtsZ is not only indispensable for cell division but is also the first protein
to localize to the division site and orchestrates the recruitment of more than
a dozen other proteins required for septation [1, 27, 28, 29, 30]. FtsZ is
also thought to be an ancient protein that originated as soon as cells formed
during an early stage of genetic code evolution [31]. Thus, understanding FtsZ
function and its assembly into the cytokinetic ring remains a fundamental
question central to biology. FtsZ, therefore, has received extensive attention
in the past decades, both in experiments [32, 33, 17, 34, 35] and in numerical
studies [36, 37, 38, 39, 40, 41, 42]. Numerical studies in two dimensions
showed that lateral attraction plays a crucial part in determining the final
shape of the FtsZ filament [43]. However, this model [43] lacked spontaneous
torsion, which, as shown recently in Ref. [44], is important in generating the
required constriction force. In recent years, considerable advancements have
been made in comprehending the role, function, and mechanisms of FtsZ,
particularly how it produces constriction force on the membrane.

The FtsZ filaments constituting rings are found to be highly dynamic [32, 17,
45, 20]. Recently, it has been shown that treadmilling not only guides the
cell wall synthesis machinery [22, 21, 46, 47, Schäper2024] but also plays a
crucial role in FtsZ ring assembly [49]. FtsZ polymerizes and forms a large
variety of dynamic structures, straight, bent, and helical filaments, which
are highly dependent on the experimental conditions such as pH, the presence
of various monovalent salts, crowding agents, and polycations [50]. Using the
structure of crystallized Methanococcus jannaschii FtsZ dimer and numerical
simulations, the torsional and bending curvatures of FtsZ filaments were
extracted [51]. The intrinsic torsional curvature present in the FtsZ could
deform the membranes via torsional stress and show chiral treadmilling
vortices in vivo [52]. Earlier experiments also suggested that GTPase activity
and concentration of free $\text{Mg}^{2+}$ ions play an important role in the
dynamics and ring formation [53, 54], and free $\text{Mg}^{2+}$ ions favor the
lateral contacts between the filaments.

While these in vivo and in vitro reconstitution experiments have generated
deep insights into the assembly dynamic of FtsZ filaments, a slightly
different approach to studying bacterial cytoskeleton assembly in a
molecularly crowded cellular environment was taken by Balasubramanian and
colleagues [19]. The authors expressed FtsZ in yeast cells, both budding yeast
(Saccharomyces cerevisiae) as well as fission yeast (Schizosaccharomyces
pombe), and showed that FtsZ can assemble into a ring-like structure even in
the absence of all other bacterial cytokinetic proteins [19]. They further
showed that the process of ring assembly involves a spooling-like mechanism,
possibly driven by GTP hydrolysis and lateral associations among the FtsZ
filaments.

In this paper, we aim to understand the mechanisms behind the emergence of
observed FtsZ filament morphologies in both bacterial and yeast systems using
a polymer model and Langevin dynamics simulations. Our theoretical approach
assumes the existence of FtsZ filamentous assembly as the starting point. We
model it as a tangentially active chiral semiflexible polymer characterized by
finite bending and torsional rigidity, as well as intrinsic bending and
torsional curvatures. A tangential activity is used to model the FtsZ
treadmilling. Our primary achievements are: $(i)$ Our model accurately
captures all experimentally observed FtsZ morphologies, such as rings,
helices, filaments, and globules, observed in FtsZ from different bacterial
species or FtsZ mutants of Escherichia coli. $(ii)$ A minimal lateral
attraction is necessary for ring formation, and treadmilling can flatten and
widen globules into rings, displaying rotation and a spooling mechanism to
ring formation. $(iii)$ The ring state remains stable over a broad range of
treadmilling activities before transitioning into the open helix (OH) through
phase coexistence. We obtain a detailed phase diagram, with the phase
boundaries marked by peaks in size fluctuations and extensile response. $(iv)$
Active spinning chiral rings (SCR) induce contractile stresses, rotation, and
shear that intensify near the phase boundaries.

The paper is organized as follows: In Sec. II, we first summarise the
experimental results, setting the motivation for the theoretical study. In
Sec. III, we present the model and simulation details. In Sec. IV, we discuss
the main results, including the phase diagram, shape and size of the polymer,
and stress generation. Finally, we conclude in Sec. V, summarizing our results
and presenting an outlook.

##  II Experimental results: motivation for theory

The various shapes of FtsZ assemblies, like rings, helices, filaments, and
condensates that have been found in several in vitro experiments and in vivo
studies in bacteria have also been observed for several kinds of FtsZs
expressed in fission yeast. Fission yeast has served as a useful model to
study the assembly dynamics of ring formation by FtsZ [19]. FtsZ from
different bacterial species or FtsZ mutants of Escherichia coli have been
shown to assemble into different forms such as helices [55] (Helicobacter
pylori FtsZ or HpFtsZ), rings [19, 55] (EcFtsZ, Staphylococcus aureus SaFtsZ
and EcFtsZ Q47K), globules (EcFtsZ D209A), open chains (EcFtsZ Q47K and EcFtsZ
Q47KD86K), closed chains (EcFtsZ Q47KD86K) and network of polymers [19]
(EcFtsZ and EcFtsZ G105S). The different forms of FtsZ assemblies that have
been thus far seen upon expression in fission yeast are shown in Fig.1($a$).
Further, since linear and helical polymers of FtsZ have been shown to undergo
a transition into rings, or vice-versa in yeast and bacterial systems [19, 23,
24], the forms observed for the FtsZ mutants in yeast cells might suggest that
the mutations drive and stabilize one or the other states. Dynamics of EcFtsZ
Q47K ring formation in fission yeast (S. pombe) from an open chain shows a
spooling mechanism.

As revealed by independent electron microscopy studies [56], the filaments
observed in fluorescent microscopy Fig. 1($a$) are typically filament bundles
composed of multiple FtsZ chains bound laterally to each other. In the
following section, we present a polymer model for such filament bundles.

Figure 1:  ($a$) Images showing the various forms of FtsZ polymers when
expressed in fission yeast. (i) ring-like structures of E. coli FtsZ (EcFtsZ),
(ii) helical or spiral polymers of Helicobacter pylori FtsZ, (iii) straight
linear polymers (open chains) of EcFtsZ Q47KD86K mutant, (iv) closed loops
(closed chains) formed by EcFtsZ Q47KD86K mutant, (v) flexible polymers or
networks formed by FtsZ G105S mutant and (vi) blobs or globules formed by the
GTPase catalytic mutant EcFtsZ D209A. The scale bar represents $3\mu m$. ($b$)
Schematic diagram of the active chains studied at different parameter regimes
and lateral attraction (LA), recapitulating conformations in ($a$). ($c$)
Schematic representation of the active chain with active force $f_{a}$ acting
tangentially along the chain’s backbone. ($d$) Spooling dynamics of ring
formation: Configurations of the active chain in the presence of lateral
attraction at $Pe=1$ and $\tilde{\epsilon}=1$ at different stages of ring
formation with time $t_{i}$. Dynamics of EcFtsZ Q47K ring formation in fission
yeast (S. pombe) from an open chain. Time-lapse imaging shows the different
stages of ring assembly (time is in minutes). In this study, all plots of the
model polymer are generated using Ovito [57], and the filament color coding
from blue to red represents the filament’s tail to head, the direction of
treadmilling.

##  III Model

We consider an active bead-spring chain with bending and torsional rigidity to
model the FtsZ filament. The polymer is denoted by a space curve described by
position vectors $\\{{\bf r}_{1},{\bf r}_{2}\,\dots,{\bf r}_{N}\\}$
corresponding to $N$ beads. The bonding energy is given by
$U_{b}=\frac{k_{S}}{2}\sum_{i=1}^{N-1}\left(|{\bf b}_{i}|-r_{0}\right)^{2}$
where the bond vector ${\bf b}_{i}={\bf r}_{i+1}-{\bf r}_{i}$, the rest-length
of the bonds is $r_{0}$, and $k_{S}$ is the bond stiffness. The bending energy
is given by
$U_{\theta}=\frac{\kappa_{\theta}}{2}\sum_{i=1}^{N-2}\left(\theta_{i}-\langle\theta\rangle\right)^{2}$,
where $\theta_{i}$ is the angle between two consecutive bond vectors ${\bf
b}_{i}$ and ${\bf b}_{i+1}$, and $\kappa_{\theta}$ denotes the bending
modulus. The non-planar nature of the space curve is described by the dihedral
angle $\delta_{i}$ between two successive planes constituted of bond-vectors
$\\{{\bf b}_{i},\leavevmode\nobreak\ {\bf b}_{i+1}\\}$ and $\\{{\bf
b}_{i+1},\leavevmode\nobreak\ {\bf b}_{i+2}\\}$ involving three successive
segments. The torsional energy of the chain is modeled by
$U_{\phi}=\frac{\kappa_{\phi}}{2}\sum_{i=1}^{N-3}\left(\phi_{i}-\langle\phi\rangle\right)^{2}$,
where $\kappa_{\phi}$ is the torsional modulus.

The self-avoidance and lateral attraction in the chain are modeled using an
interaction potential $U_{NB}$ between non-bonded monomers, with

$U_{\textrm{\text{NB}}}(r)=\begin{cases}\sum_{\langle
i,j\rangle}4\epsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right]+\epsilon&\text{for}\
r_{ij}\leq r_{\text{min}}\cr 0&\text{for}\ r_{ij}>r_{\text{min}}.\end{cases}$
(1)

A choice of $r_{\text{min}}=2^{1/6}\sigma$ makes the above potential purely
repulsive, the Weeks-Chandler-Andersen (WCA) interaction. This choice can
model a self-avoidance. In addition, attraction between filament segments can
be incorporated by replacing the distance cutoff by the typical Lennard-Jones
(LJ) choice $r_{\text{min}}=2.5\sigma$ so that monomers at a separation beyond
the WCA cutoff attract each other via the LJ potential. The total energy cost
is $U=U_{\textrm{b}}+U_{\theta}+U_{\phi}+U_{\textrm{NB}}$.

We model the treadmilling activity in terms of a tangential active force
$f_{a}{\bf b}_{i}$ that acts along each bond. Here $f_{a}$ denotes a force per
unit length [58]. This leads to a total self-propulsion force ${\bf
F}_{a}=\sum_{i=1}^{N-1}f_{a}{\bf b}_{i}$. We use the dimensionless Péclet
number,

$Pe=\frac{f_{a}r_{0}^{2}}{k_{B}T},$ (2)

to characterize the local activity. Note that, for a chain of length
$L=(N-1)r_{0}$, a related measure of activity $Pe_{L}=Pe(L/r_{0})^{2}$ that
scales quadratically with the chain length has been used in recent studies of
active polymers [58].

The Langevin dynamics of the chain are described by

$m\dot{\bm{v}}_{i}=-{\bm{\nabla}}_{i}U-\gamma{\bm{v}}_{i}+\gamma\sqrt{2D}\,{\bm{\eta}}_{i}+{\bm{f}}_{a}^{i}.$
(3)

where, ${\bm{f}}_{a}^{i}=f_{a}{\bf u}_{i}$ with ${\bf u}_{i}=({\bf
b}_{i-1}+{\bf b}_{i})/2$ for $i=2,\dots,(N-1)$ and ${\bf u}_{1}={\bf
b}_{1}/2$, ${\bf u}_{N}={\bf b}_{N-1}/2$. In the absence of active drive, the
above equation obeys equilibrium fluctuation-dissipation with monomer
diffusivity $D=k_{B}T/\gamma$, and Gaussian random noise ${\bm{\eta}}_{i}$
satisfying $\langle{\bm{\eta}}_{i}\rangle=0$ and
$\langle{\bm{\eta}}_{i}(t)\otimes{\bm{\eta}}_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})\mathbb{1}$.
The length, energy, and time scales are set by $\sigma=r_{0}$, $k_{B}T$, and
$\tau=\sigma\sqrt{m/k_{B}T}$, where $\tau$ also describes the time scale for
diffusion over $\sigma$. In these units, we choose $k_{S}=100$ and set
$\kappa_{\theta}/\kappa_{\phi}=2.915$, $\langle\theta\rangle=7.6^{\circ}$ and
$\langle\phi\rangle=20^{\circ}$ in accordance with previous all-atom
simulations that were in agreement with properties of wild-type FtsZ filaments
[51]. Further, we choose $\kappa_{\theta}=20$, such that the ratio of the
chain length $L$ to the persistence length $\ell_{p}$ of the polymer,
$L/\ell_{p}=5$, consistent with measured persistence lengths [19] (also see
Appendix-A). The strength of the inter-segment attraction is tuned using the
dimensionless control parameter $\tilde{\epsilon}=\epsilon/k_{B}T$. The
Langevin dynamics simulations are performed using $\delta t=10^{-4}\,\tau$
within the velocity-Verlet implementation in LAMMPS [59] with in-house
modifications to incorporate the tangential activity.

##  IV Results

FtsZ filaments exhibit a variety of structural forms, including helices,
rings, open filaments, and globules, as shown in Fig.1($a$) and captured
within our model, Fig.1($b$). In the absence of lateral attraction (WCA
repulsion between segments) and activity, the model proposed above leads to a
native state of equilibrium helical configurations [51]. This conformation can
be seen in the top panel of Fig.1($b$). Such morphologies can relax to simple
equilibrium open chains in the absence of helicity in two limits of (i)
vanishing torsional rigidity or (ii) intrinsic curvature
$\langle\phi\rangle=\pi$ constraining polymeric fluctuations to an arbitrary
plane. Examples of such conformations are shown in the bottom right panels in
Fig.1($b$). As mentioned before, Fig.1($c$) depicts the tangential activity
implemented in our model.

On the other hand, large lateral attraction collapses the equilibrium filament
into a compact globule. As mentioned before, the attraction between polymer
segments is incorporated within our model by the full Lennard-Jones potential.
This is similar to the well-known coil-globule transition [60]. However, the
globule, in this case, comprises a compact arrangement of non-coplanar rings
lying in different orientations. In contrast, at an intermediate lateral
attraction, the open helix first transitions into a ring. In the presence of
activity, the ring rotates in a chiral fashion to form spinning chiral rings
(SCR). These dynamics distinguish them from the equilibrium rings and, as we
show later, produce active contractile stress. As depicted in Fig. 1($d$) and
Supplemental Movie 1, the active helical chain at $Pe=1$ and
$\tilde{\epsilon}=1$ transitions from an open helix conformation to SCR
through a spooling-like mechanism capturing experimental findings, also
observed before by one of us in Ref. [19].

Figure 2: $(a)$ The dependence of the radius of gyration $\langle
r_{g}\rangle$ and prolateness $\langle p\rangle$ on lateral attraction
strength $\tilde{\epsilon}$ at equilibrium ($Pe=0$). The system exhibits an
open conformation for low $\tilde{\epsilon}$, transitioning to ring and
globule conformations for higher $\tilde{\epsilon}$. $(b)$ The behavior of the
principal moments (eigenvalues) of the gyration tensor, $\lambda_{i}$’s, with
respect to $\tilde{\epsilon}$. $(c)$ Representation of typical equilibrium
configurations for different values of lateral attraction strength
($\tilde{\epsilon}$). The system parameters are set as follows: bending
modulus $\kappa_{\theta}=20$, intrinsic bending curvature
$\langle\theta\rangle=7.6^{\circ}$, and intrinsic torsional curvature
$\langle\phi\rangle=20^{\circ}$.

###  IV.1 Equilibrium helix to ring and globule

With increasing inter-segment attraction $\tilde{\epsilon}=\epsilon/k_{B}T$,
the open helix state of the polymer first undergoes a transition to a ring. At
further higher attraction, the single ring further collapses into a globule
consisting of multiple internal ring-like conformations of different
orientations; see Fig.2. The transitions are quantified utilizing the radius
of gyration tensor [60, 61, 62, 63]

$\displaystyle\mathcal{R}_{\alpha\beta}=\frac{1}{2N^{2}}\sum_{i,j=1}^{N}(r_{i\alpha}-r_{j\alpha})(r_{i\beta}-r_{j\beta})$
(4)

for a chain of $N$ beads with $r_{i\alpha}$ denoting the $\alpha$-th component
of the position vector ${{\bf r}}_{i}$ of $i$-th bead. With
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$ denoting the three eigenvalues of
$R_{\alpha\beta}$, the principal radii of gyration, and using

$\displaystyle\lambda_{tr}=\lambda_{1}+\lambda_{2}+\lambda_{3},$ (5)

we can estimate the mean polymer size as $\langle
r_{g}\rangle=\sqrt{\langle\lambda_{tr}\rangle}$, and use the prolateness

$\displaystyle
p=\frac{\prod_{i=1}^{3}(3\lambda_{i}-\lambda_{tr})}{\lambda^{3}_{tr}},$ (6)

to obtain the mean prolateness $\langle p\rangle$ averaging over all steady-
state configurations. A related positive quantity, asphericity, which vanishes
for a spherically symmetric shape, is utilized later in this paper. The mean
prolateness $\langle p\rangle$ is bounded above and below: $-1/4\,\leq\langle
p\rangle\leq 2$, with positive (negative) values of $p$ denoting prolate
(oblate) shapes, and $\langle p\rangle=0$ for isotropic conformations.

The polymer size shows a sharp drop across the transition of an open helix to
an equilibrium ring at $\tilde{\epsilon}=\tilde{\epsilon}^{\ast}$; see
Fig.2($a$). At higher $\tilde{\epsilon}$, the size $r_{g}$ keeps on
decreasing, although ever so little. This decrease in size is associated with
another structural ‘transition’, from ring to compact globule. This transition
is further quantified in terms of the prolateness $\langle p\rangle$, which
grows from negative values ($\langle p\rangle=-0.1$ at
$\tilde{\epsilon}^{\ast}$) for the ring to vanish for the spherically
symmetric globule. Note that, here, the ring state cannot be a two-dimensional
object but must have a height due to the width of the filament and self-
avoidance. Thus, the ring state has approximately the shape of a short pillbox
with diameter $\lambda_{1}=\lambda_{2}\equiv\lambda$ and height $\lambda_{<}$.
Within this approximation, one gets

$\displaystyle\langle
p\rangle=-\frac{1}{4}\left\langle\left(\frac{1-\frac{\lambda_{<}}{\lambda}}{1+\frac{\lambda_{<}}{2\lambda}}\right)^{3}\right\rangle\approx-\frac{1}{4}\left(1-\left\langle\frac{9\lambda_{<}}{2\lambda}\right\rangle\right),$
(7)

linearizing for small parameter $\langle\lambda_{<}/\lambda\rangle$. Thus, in
the limit of vanishing height with respect to the ring size,
$\langle\lambda_{<}/\lambda\rangle\to 0$, one expects $p\approx-0.25$. The
numerical simulation results in Fig.2($a$) show the actual $p$ value at the
transition point remains above this estimate, suggesting a non-zero
$\langle\lambda_{<}/\lambda\rangle\approx 2/15$, an estimate that agrees
approximately with $\langle\lambda_{3}/\lambda_{1}\rangle$ at
$\tilde{\epsilon}^{\ast}$ shown in Fig.2($b$). With increasing
$\tilde{\epsilon}$ beyond this transition point, $\langle p\rangle$ increases
as the ring transitions towards a globule to vanish at $\tilde{\epsilon}=3$,
identifying the spherical symmetry of the final globule state.

Figure 3: $(a)$ Configurational phase diagram in the $Pe-\tilde{\epsilon}$
plane. The active chains are color-coded from the tail (blue) to the head
(red), with self-propulsion directed from the tail to the head. Note that in
terms of an alternative measure $Pe\,(L/r_{0})^{2}$, the polymer activity in
the figure ranges from $0$ to $2\times 10^{4}$. $(b)$ Quantitative phase
diagram in the $Pe-\epsilon$ plane using fluctuations
$\sigma^{2}=\langle\tilde{r}_{e}^{2}\rangle-\langle\tilde{r}_{e}\rangle^{2}$
of the scaled end-to-end separation $\tilde{r}_{e}=r_{e}/L$ represented by
points denoted by colors as shown in the color bar. $(c)$ Plot of $\sigma^{2}$
as a function of $Pe$ for a fixed $\tilde{\epsilon}=1$ and shows a maximum at
$Pe^{*}=0.5$ and decreases away from this point. The plot is shown by blue
($\diamond$) data points and the axis label on the right. The extension
modulus $\chi$ is plotted with the red-colored left axis and red ($\triangle$)
data points. $\chi$ exhibits a non-monotonic increase with activity $Pe$ and
peaks near the transition point $Pe^{**}=0.1$. The green-colored symbol
denotes a representative error bar. $(d)$ Probability distributions
$p(\tilde{r}_{e})$ for various $Pe$ values at $\tilde{\epsilon}=2$. The clear
bimodality in the distribution function at $Pe=1$ shows the coexistence of
compact globule and ring structures with open helical conformation. Inset:
compares the distributions at $Pe=0$ and $10^{-3}$. $(e)$ Probability
distributions $P(p)$ of the prolateness $p$ for the same $Pe$ values as in
$(d)$ and $\tilde{\epsilon}=2$. At $Pe=1$, the distribution shows maxima
corresponding to globule (peak around $p=0$), ring (peaks at $p\approx-0.1$
and $-0.15$), and open helix (peak for $p\approx 1.5$).

These transitions can be directly read off from the variation of
$\lambda_{i}$s in Fig.2($b$). In the open helix state, all the eigenvalues are
different from each other. At the formation of the spiral, two larger
eigenvalues corresponding to the diameter of the ring become equal,
$\lambda_{1}=\lambda_{2}$, while the height of this pillbox-shape
$\lambda_{3}$ remains smaller. With increasing $\tilde{\epsilon}$,
$\lambda_{1}=\lambda_{2}$ shrinks, and the $\lambda_{3}$ increases to become
equal at the formation of spherical globule at $\tilde{\epsilon}=3$. Note that
the transformation of the ring state to the globule is gradual and does not
show any signature of phase transition. As we show in the later section, the
presence of treadmilling activity can transform a globule into a ring, thereby
potentially stabilizing rings in the biological context.

###  IV.2 Role of treadmilling activity

The shape of the equilibrium ring formed at $\tilde{\epsilon}\approx 1$
remains intact over a wide range of treadmilling activity $Pe$. Before opening
up, the ring shows chiral rotation, characterizing the non-equilibrium SCR
state. However, for higher $\tilde{\epsilon}$ that stabilize globular
structures at equilibrium, activity helps to unwind the globule to form
rotating rings, thereby stabilizing SCR. At even higher $Pe$, the rings do
unfold into open helices. For example, at $\tilde{\epsilon}\approx 1$, the
ring opens up into a helix above $Pe\approx 0.1$.

We present a comprehensive phase diagram in Fig. 3$(a)$ and ($b$) as a
function of treadmilling activity $Pe$ and lateral attraction strength
$\tilde{\epsilon}$. Fig. 3$(a)$ illustrates representative conformations of
the chain at different parameter values. This plot shows representative
conformations at $\tilde{\epsilon}\geq 0.8$ where the equilibrium chain is
already in a ring or globule state. For a similar but more detailed phase
diagram, see Appendix- B. At any given $\tilde{\epsilon}$, the ring
destabilizes at high enough $Pe$ to form an open helix. At small to
intermediate $\tilde{\epsilon}$, the figure suggests that the ring state
remains stable over a broad range of $Pe$. However, at high enough
$\tilde{\epsilon}$ (e.g., at $\tilde{\epsilon}\approx 2.4$), the equilibrium
chain is in a globule state. With increasing $Pe$, the globule first
transitions to SCR before undergoing a transition to open helix at high enough
$Pe$ ($\approx 1$) as shown in Fig. 4

Figure 4: The mean prolateness $\langle p\rangle$ as a function of activity
$Pe$ for various $\tilde{\epsilon}$ values. The globule formed at equilibrium
condition ($Pe=0$) transitions into a ring shape as a function of activity
($Pe$). Solid lines are guides to the eye. Inset: Variation of principal radii
of gyration, $\lambda_{i}$’s, with $Pe$ at $\tilde{\epsilon}=2.4$, with
symbols for $i=1$ ($\ast$), $i=2$ ($\times$) and $i=3$ ($+$).

In Fig. 3 (b), we present a quantitative version of the phase diagram using
fluctuations in the chain’s end-to-end separation. The color bar indicates the
amount of fluctuations
$\sigma^{2}=\langle\tilde{r}_{e}^{2}\rangle-\langle\tilde{r}_{e}\rangle^{2}$
in the scaled end-to-end separation $\tilde{r}_{e}=r_{e}/L$, where $L$ is the
chain length. Near the transition from ring to open helix denoted by the
dashed red line in Fig. 3(b), $\sigma^{2}$ reaches a maximum. Fig. 3(c) shows
the variation of $\sigma^{2}$ as a function of $Pe$ for a fixed $\epsilon=1$
(blue colored points and ordinate label) with the maximum at $Pe^{*}=0.5$, the
transition point. It sharply diminishes away from this transition point.

This figure also shows the linear response $\chi$ of the polymer extension
under external force (see Appendix-C and Fig.9). It varies non-monotonically
with $Pe$ and shows a maximum that precedes the maximum in $\sigma^{2}$, e.g.,
at $Pe^{**}=0.1$ for $\epsilon=1$. The maximum is attributable to the
instability of the ring structure near the transition, where even a small
amount of extensile force is sufficient to destabilize the ring and open it.

Figure 5: Fourier transform $S_{{\bf t}{\bf t}}(q)$ of the tangent-tangent
correlation function $C_{{\bf t}{\bf t}}(s)$ for the helical chain for various
$Pe$ values at fixed lateral attraction $\tilde{\epsilon}=1.0$. The
correlation function is oscillatory for the ring phase, with the periodicity
captured by the peak in the structure factor. The peak height quantifies the
degree of helicity of the chain, while the width of the peak indicates the
statistical dispersion of the structure.

In Fig. 3$(d)$ and $(e)$, we present the probability distributions
$p(\tilde{r}_{e})$ and prolateness $p$ respectively, at various $Pe$ keeping
$\tilde{\epsilon}=2$ fixed. The distribution functions are normalized to
$\int_{0}^{1}d\tilde{r}_{e}\,4\pi\tilde{r}_{e}^{2}\,P(\tilde{r}_{e})=1$. At
$Pe=0$ and $10^{-3}$, where the ring phase is stable, the distribution shows a
single peak near $\tilde{r}_{e}\approx 0.03$; see Fig. 3$(d)$ and its inset.
The compact structure at equilibrium swells under activity $Pe=10^{-3}$ to
become more ring-like. Near the transition point, $Pe=1$, the chain shows
bimodality in $P(\tilde{r}_{e})$, indicating the coexistence of open helix
(maximum near $\tilde{r}_{e}=0.45$) and compact ring (maximum near
$\tilde{r}_{e}\lesssim 0.01$). At higher $Pe$, the distribution becomes
unimodal with a maximum around $\tilde{r}_{e}=0.45$ for $Pe=2$, corresponding
to the open helix. The broad distribution characterizes large deviations
around this maximum, allowing the chain to visit other conformations.

The distribution of prolateness $p$ remains unimodal at $Pe=0$ and $10^{-3}$
(Fig. 3$(e)$ and inset). The location of the maximum shifts towards smaller
$p$ values at $Pe=10^{-3}$, with respect to equilibrium. This is due to the
reorganization to a better ring via the non-equilibrium swelling. At $Pe=1$,
the multimodal distribution function $P(p)$ displays the coexistence of ring,
globule, and open helix, captured by the peaks at $p<0$, $p\approx 0$, and
$p>0$, respectively. Moreover, at this Péclet, the distribution shows a second
maximum with split peaks at $p<0$. The first one near $p=-0.1$ corresponds to
the stable ring phase. The second one at $p<-0.1$ corresponds to a more
swollen ring that is formed on the way of opening up from the ring state to
the open helical conformation. The reduction of $p$ with swelling can be
understood using larger $\lambda$ in Eq.(7). At a higher activity, $Pe=2$, as
the chain opens up, the distribution becomes unimodal again with one maximum
near $p\approx 1.25$.

The variation of mean prolateness $\langle p\rangle$ with $Pe$ corresponding
to different $\tilde{\epsilon}$ is shown in Fig.4. The equilibrium ring state
at $\tilde{\epsilon}=1$ remains relatively unaltered up to $Pe\approx 0.1$,
where the ring transitions into an open helix. On the other hand, $\langle
p\rangle$ increases with $\tilde{\epsilon}$ at $Pe=0$ to vanish at large
$\tilde{\epsilon}$, as the equilibrium ring gets into globule-like shape; see
Fig.2 as well. With increasing $Pe$, the equilibrium globules at
$\tilde{\epsilon}=2$ and $2.4$ become ring-like, showing a decrease in
$\langle p\rangle$ reaching a negative minimum at an intermediate value of
$Pe$ ($Pe=10^{-3}$) before increasing with activity as the rings open up. This
suggests that treadmilling activity can regulate the formation of FtsZ rings.
This feature is further characterized using the three principal radii of
gyration $\langle\lambda_{i}\rangle^{1/2}$ with $i=1,2,3$, for
$\tilde{\epsilon}=2.4$ shown in the inset of Fig.4. At $Pe=10^{-3}$,
$\langle\lambda_{1}\rangle$ and $\langle\lambda_{2}\rangle$ becomes largest as
the ring widens and associated with this $\langle\lambda_{3}\rangle$ gets to
the smallest value signifying the maximal thinning of the ring.

The change in mean size and shape of the active chiral chain can further be
analyzed in terms of the mean radius of gyration, asphericity, and
prolateness. For example, see Appendix-D and Fig.10.

The number of turns in the SCR does not change before it opens up; however,
the robustness of the ring state varies non-monotonically. This is established
using the tangent-tangent correlation function $C_{{\bf t}{\bf
t}}(s)=\langle{\bf t}(s)\cdot{\bf t}(0)$ along the chain contour, and its
Fourier transforms $S_{{\bf t}{\bf t}}(q)$; see Fig.5. The correlation
function shows periodic variation with the periodicity captured by the
pronounced maximum in $S_{{\bf t}{\bf t}}(q)$ with the location of the maximum
near $q=n/L$ denoting the number of turns $n$. Fig.5 shows the maximum is at
$n=qL\approx 6$. As we see, this position of maximum does not change with
$Pe$; however, the height of $S_{{\bf t}{\bf t}}(q)$ at the maximum varies
non-monotonically. The initial increase with $Pe$ characterizes the formation
of a more stable ring. The subsequent decrease of this peak height at even
larger $Pe$ signifies the destabilization of the ring.

####  IV.2.1 Dynamics near the SCR-open helix phase boundary

The dynamics of the chain near the spiral to open helix transition can be
analyzed using the time series of the principal radii of gyration
$\lambda_{i}$ with $i=1,2,3$ and the prolateness $p$; see Fig. 6. The time
series in Fig. 6($a$) shows signatures of a coexisting compact ring (denoted
by $R1$), swelled ring ($R2$), globule ($G$), and open helix ($OH$). The
globule is characterized by $\lambda_{1}=\lambda_{2}=\lambda_{3}$ and $p=0$.
The data at $G$ shows approximate agreement with this expectation. In the ring
state, $\lambda_{1}\approx\lambda_{2}$ is significantly larger than
$\lambda_{3}$ and $p(R2)<p(R1)$ with both R1 and R2 rings having negative $p$.
For OH, $\lambda_{1}\gg\lambda_{2}(\approx\lambda_{3})$ and $p\to 2$ (also see
Fig.11 in Appendix-E).

Figure 6: $(a)$ A representative time series of $p$ and the principal radii of
gyration at $Pe=1$ and $\tilde{\epsilon}=2$ where ring (marked with $R1$,
$R2$), globule ($G$) and open helix ($OH$) co-exist. Here $R1$ and $R2$
represent the compact and swollen ring states with $p=-0.1$ and $p=-0.15$,
respectively. $(b)$ Two-time auto-correlation function of $p$ for various $Pe$
values at $\tilde{\epsilon}=2$. $(c)$ Power spectral densities $S_{pp}(\nu)$
for the same $\tilde{\epsilon}$ and $Pe$ values as in $(b)$. The scatter plots
denote the direct Fourier transform, and the solid lines are guides to the
eye. The black dashed, dash-dotted and dotted lines represent $\nu^{-1}$,
$\nu^{-2}$ and $\nu^{-3}$, respectively.

These dynamics can be further characterized in terms of two-time correlations
and the corresponding power spectral density. Fig. 6($b$) shows the
correlation function $C_{pp}(t)=\langle\delta p(t)\delta
p(0)\rangle/\langle\delta p^{2}(0)\rangle$ with $\delta p(t)=p(t)-\langle
p\rangle$. The corresponding power spectrum $S_{pp}(\nu)$ is obtained by
Fourier transforming the correlation function and is plotted in Fig. 6($c$).
We used TISEAN [64] to compute the correlation and power spectrum. The
apparent oscillations in $C_{pp}(t)$ at $Pe=1$ correspond to the phase
coexistence. The absence of any characteristic maximum in frequency $\nu$ in
the power spectrum $S_{pp}(\nu)$ characterizes the absence of any robust
oscillation in the dynamics. The high frequency behavior of $S_{pp}$ shows
approximate power-law decay as $\nu^{-\beta}$ with $\beta=1$ in the stable SCR
phase ($Pe=0$ and $10^{-3}$), $\beta=2$ for $Pe=1$ near the transition, and
$\beta=3$ for $Pe=2$ corresponding to open helix.

####  IV.2.2 Stress generation by the spinning chiral ring

Here, we present a direct measure of the active stress due to the spinning
chiral ring (SCR). Recent advances led to derivations of swim pressure for
active Brownian particles [65, 66, 67, 68]. Instead, we employ an embedding
environment to calculate the stress due to the active chain.

We consider a harmonic cubic lattice of size $N_{g}^{3}$ with fixed boundaries
to embed the SCR. The lattice spacing is chosen to be $\sigma$ and $N_{g}=17$
such that the box size $N_{g}\sigma$ is significantly larger than the diameter
of the SCR so that the distortions do not reach the boundaries. The monomers
of the chain are assumed to be bonded with lattice points within the cutoff
distance $\sigma$. To avoid introducing new parameters, we choose these bond
strengths and that of the harmonic lattice to be $k_{S}$, the same as in the
active polymer.

We calculate the excess stress on the cubic lattice due to the active chiral
chain

$\displaystyle\Sigma_{\alpha\beta}=-\frac{1}{{\cal
N}\sigma^{3}}\sum_{i=1}^{\cal N}\langle r_{i\alpha}f_{i\beta}\rangle\,,$ (8)

where ${\cal N}$ denotes the number of lattice points bonded with the chain,
and ${\cal N}\sigma^{3}$ is the corresponding volume occupied by them. Here,
$f_{i\beta}$ stands for the $\beta$-th component of the force acting on the
lattice point $i$ due to the chain.

Figure 7: Deviations of excess pressure $\delta\varPi^{ex}$, magnitude of the
largest eigenvalue of shear $\delta\lambda_{\tau}$, and the largest eigenvalue
of rotation $\delta\lambda_{R}$ from their equilibrium values
$\varPi^{ex}|_{eq}=0.8734$, $\lambda_{\tau}|_{eq}=0.1757$ and
$\lambda_{R}|_{eq}=0.0034$. The black dashed line represents linear growth
$\sim Pe$. Solid and dashed lines serve as visual guides, with dashed lines
indicating the $Pe$ range in which the free chain opens up into a helix.

The stress tensor $\Sigma_{\alpha\beta}$ is fully characterized by its three
irreducible parts, the excess pressure $\varPi^{ex}$, shear
$\tau_{\alpha\beta}$, and rotation $R_{\alpha\beta}$. They are expressed as

$\displaystyle\varPi^{ex}$ $\displaystyle=$
$\displaystyle-\frac{1}{d}\Sigma_{\alpha\alpha}$
$\displaystyle\tau_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\Sigma_{\alpha\beta}+\Sigma_{\beta\alpha})-\frac{1}{d}\Sigma_{\alpha\alpha}\delta_{\alpha\beta}$
$\displaystyle R_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\Sigma_{\alpha\beta}-\Sigma_{\beta\alpha}),$ (9)

where $d$ is the embedding dimension. The Einstein summation convention is
assumed above. The pressure is a scalar, $\tau_{\alpha\beta}$ is a traceless
symmetric matrix, and $R_{\alpha\beta}$ is a purely antisymmetric matrix. In
our study $d=3$. Thus, in 3d, shear $\tau_{\alpha\beta}$ has five independent
components, and rotation $R_{\alpha\beta}$ has three.

In Fig. 7, we show the dependence of the $\varPi^{ex}$, the magnitude of the
largest eigenvalues $\lambda_{\tau}$ of $\tau_{\alpha\beta}$ and the largest
eigenvalue $\lambda_{R}$ of $R_{\alpha\beta}$ as a function of activity. We
show the deviations of these quantities from equilibrium values (at $Pe=0$)
with increasing $Pe$. The deviation in excess pressure $\delta\varPi^{ex}$ is
negative, showing that the activity of the chain generates contractility, as
expected from experimental observations. The deviations in shear
$\delta\lambda_{\tau}$ and rotation $\delta\lambda_{R}$ show similar
dependence on activity; see Fig. 7. For activity higher than $Pe=1$, at which
the SCR is unstable towards the open helix, all the above measures show a
linear increase with $Pe$.

Figure 8: Configurational phase diagram in the $Pe-\tilde{\epsilon}$ plane.
The system exhibits various structures across different ranges of activity
$Pe$ and lateral attraction strength $\tilde{\epsilon}$. The chain maintains a
stable helical structure for very low values of $\tilde{\epsilon}$ and forms a
ring at intermediate $\tilde{\epsilon}$ values across a broad range of $Pe$.
At higher $\tilde{\epsilon}$ and lower $Pe$ values, the chain adopts globule
structures. Increased $Pe$ tends to favor an open chain conformation by
disrupting contacts between chain segments. Violet square boxes denote the
coexistence of ring, globule, and open helical structures forming near
transition points, where the chain spends maximum time in an open conformation
and also forms structures like rings or globules. Note that in terms of
another measure of activity defined earlier in Sec. III, $Pe$ ranges from
$10^{-2}$ to $2\times 10^{4}$.

##  V Conclusions

Our study addresses the pivotal question of how treadmilling and lateral
attraction contribute to the formation of the FtsZ filament’s ring structure
during cell division. Using a coarse-grained theoretical model that treats
FtsZ filaments as tangentially active chiral semiflexible polymers, we
accurately reproduced various morphological phases observed experimentally,
including rings, helices, filaments, and globules. While lateral attraction
stabilizes the ring structure, it fails to induce the observed net rotation
without treadmilling activity. We presented a detailed phase diagram exploring
ring stability as a function of activity and attraction, revealing a stable
ring state over a broad range of treadmilling activity before transitioning
into an open helix through phase coexistence, indicative of a first-order
phase transition. In fact, treadmilling flattens and widens otherwise globule
states to stabilize SCR and generates contractile stresses in their
surroundings. The transition from the stable ring state to the open helix
state is marked by increased size fluctuations and extensile response. While
our model treats FtsZ as an assembled polymer, we recognize that the Z-ring’s
structure is more intricate. It forms from multiple shorter treadmilling
filaments near a curved membrane surface, merging to create the Z-ring. Models
that capture this initial assembly, incorporating treadmilling rates, and
account for filament ordering on a curved membrane could offer further
insights into bacterial cytokinetic ring formation. Our findings, nonetheless,
enhance the understanding of FtsZ ring dynamics and provide a basis for
further study into bacterial cytokinesis regulation and the evolutionary link
between FtsZ protofilaments and microtubule structures.

##  VI  Acknowledgement

The numerical simulations were performed using SAMKHYA, the High-Performance
Computing Facility provided by the Institute of Physics, Bhubaneswar. Image
credits: Authors acknowledge Ajay Kumar Sharma and Sakshi Mahesh Poddar for
FtsZ images and Mirza Salim Beg at the Center for Interdisciplinary Sciences
at NISER, Jatni. We thank Amir Shee and Subhashree S Khuntia for the FtsZ
persistence length analysis. DC acknowledges a research grant from the
Department of Atomic Energy (DAE), India (1603/2/2020/IoP/R&D-II/150288) and
thanks the International Centre for Theoretical Sciences (ICTS-TIFR),
Bangalore, for an Associateship. RS acknowledges funding support from NISER,
DAE.

##  Appendix A Persistence length of open chains

We used movies of the FtsZ mutant EcFtsZ Q47KD86K that remains in open chain
conformation to extract effective persistence length $\ell_{p}$ by evaluating
tangent-tangent correlation and fitting with
$\langle\hat{u}(s)\cdot\hat{u}(0)\rangle=\exp(-s/\ell_{p})$ over shorter
length scales. We used FIJI to extract the chain conformation on which we
performed the analysis. The correlation over longer separations is known to
deviate from exponential to power-law decay due to, e.g., self-avoidance.

data sets | $L$ | $\ell_{p}$ | $\frac{\Delta\ell_{p}}{\ell_{p}}$ | $L/\ell_{p}$
---|---|---|---|---
1 | 250 | 63.39 | 3.93% | 3.94
2 | 290 | 60.84 | 2.57% | 4.76
3 | 376 | 65.26 | 2.42% | 5.76
Table 1: The values of $L$ and $\ell_{p}$ listed in the table are in units of
pixels where each pixel has a size $0.0346\,\mu$m. The length of the chains
are $L\sim 10\,\mu$m, and vary between $8\,\mu$m to $13\,\mu$m. The effective
persistence length $\ell_{p}\approx 2\,\mu$m, equivalent to wild type FtsZ
values measured earlier [19].

##  Appendix B Configurational phase diagram

The main text presents a configurational phase diagram for a few selected
parameter values in the $Pe-\tilde{\epsilon}$ plane. In Fig. 8, we show a full
conformational phase diagram over a wider range of parameter values.

Figure 9: Extension $\delta x_{e}$ as a function of external force $f_{s}$ for
various values of $Pe$ and fixed $\tilde{\epsilon}=1$. The data points are
color-coded: orange for the ring phase, turquoise for the open helix
conformations, and red indicating the transition point ($Pe=0.1$). Dotted
lines depict linear fits to the data for various $Pe$ with their respective
slope values.

##  Appendix C Extensile response

In Fig. 9, we show the change in the end-to-end distance along the
$x$-direction with respect to the steady state distance $\delta x_{e}$
($:=\langle x_{e}\rangle_{f_{s}}-\langle x_{e}\rangle_{f_{s}=0}$) as a
function of the applied stretching force $f_{s}$ at different $Pe$ and fixed
$\tilde{\epsilon}=1$. For all $Pe$ values, $\delta x_{e}$ scale linearly with
small $f_{s}$ as shown with the dotted lines in Fig. 9. The linear fits are
used to calculate the extensile linear response $\chi=\frac{\partial\delta
x_{e}}{\delta f_{s}}$ used in the main text. Note that because of self-
avoidance, compression of the rings that already have a compact structure is
strongly resisted, and the slope of the response sharply changes for negative
$f_{s}$.

##  Appendix D Size and shape of the polymer chain

Figure 10: Radius of gyration ($r_{g}$), asphericity ($\mathfrak{a}$),
thickness ($\sqrt{\lambda_{3}}$); smallest eigenvalue of the radius of
gyration tensor) and prolateness ($p$) as function of $Pe$ for fixed lateral
attraction $\tilde{\epsilon}=1$ ($a$) and as a function of $\tilde{\epsilon}$
for fixed $Pe=10^{-2}$ ($b$). In panel ($a$), radius of gyration is scaled by
$r_{g}^{(max)}=14.39$ and thickness by $\sqrt{\lambda_{3}}^{(max)}=2.4$ that
are the value of $r_{g}$ and $\sqrt{\lambda_{3}}$, respectivley, for $Pe=2$.

As shown in the main text, the size and shape of a polymer can be quantified
using the eigenvalues of the radius of gyration tensor, specifically using the
radius of gyration $r_{g}$ (Eq. 5) and prolateness $p$ (Eq. 6). In addition,
here we use the asphericity $\mathfrak{a}$ of a configuration, given by

$\displaystyle\mathfrak{a}=\frac{3}{2}\frac{\sum_{i=1}^{3}(\lambda_{i}-\frac{\lambda_{tr}}{3})^{2}}{\lambda^{2}_{tr}}=1-3\frac{\lambda_{d}}{\lambda^{2}_{tr}}$
(10)

where
$\lambda_{d}=\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}+\lambda_{3}\lambda_{1}$.
The mean asphericity $\langle\mathfrak{a}\rangle$ is obtained by averaging
over steady-state configurations. For a perfect isotropy
$\langle\mathfrak{a}\rangle=0$ and $\langle p\rangle=0$. The asphericity
$\mathfrak{a}$ is bounded above and below $0\leq\mathfrak{a}\leq 1$.

In Fig. 10$(a)$, we show the variation of all these parameters as activity
increases at $\tilde{\epsilon}=1$. For a wide range of activities (
$10^{-6}\leq Pe\leq 10^{-1}$) rings are stable with $\langle r_{g}\rangle$,
$\langle p\rangle<0$, $\langle\mathfrak{a}\rangle>0$ remaining unchanged. Note
that $\langle\lambda_{3}\rangle$ is a measure of ring thickness in the SCR
phase. This also remains unchanged up to $Pe=0.1$. At high enough activity,
$Pe>0.1$, intersegment attraction gets overpowered by activity to break
contacts between the monomers and destabilize the ring to open a helix state.
Examples of the equilibrium ring and active rotating SCR are shown in
Supplemental Movie 2. In Fig. 10$(b)$, we show the variation of the same size
and shape parameters as a function of $\tilde{\epsilon}$ at a fixed
$Pe=10^{-2}$. Increasing $\tilde{\epsilon}$ shrinks the size $\langle
r_{g}\rangle$. A clear crossover from ring to globule is captured by the
vanishing of $\langle p\rangle$ and $\langle\mathfrak{a}\rangle$.

Figure 11: $(a)$ Full time series of prolateness $p$ at $Pe=1$ and
$\tilde{\epsilon}=2$ where ring (marked with $R1$, $R2$), globule ($G$) and
open helix ($OH$) co-exist. The dashed magenta line corresponds to $S=0$.
$(b)$, $(c)$ are two-time auto-correlation functions of radius of gyration
($r_{g}$) and asphericity ($\mathfrak{a}$), respectively, for various $Pe$
values at $\tilde{\epsilon}=2$.

##  Appendix E Dynamics: Time-series and correlation

Here, we show a long-time series of prolateness $-1/4\leq p\leq 2$ near a
transition point $\tilde{\epsilon}=2$ and $Pe=1$. The value of $p$ shows large
excursions between open helix (large positive $p$), globule $p=0$, and ring
$p<0$ states; see Fig.11($a$). The correlation functions
$C_{rr}(t)=\langle\delta r_{g}(t)\delta r_{g}(0)\rangle/\langle\delta
r_{g}^{2}(0)\rangle$ for the radius of gyration $r_{g}(t)$ and
$C_{\mathfrak{a}\mathfrak{a}}(t)=\langle\delta\mathfrak{a}(t)\delta\mathfrak{a}(0)\rangle/\langle\delta\mathfrak{a}^{2}(0)\rangle$
asphericity $\mathfrak{a}(t)$ show similar features as that of $C_{pp}(t)$
shown in the main text. The apparent oscillations in the correlations at
$Pe=1$ do not correspond to any specific oscillatory Fourier mode, as verified
using the corresponding power spectra having features similar to $S_{pp}(\nu)$
shown in the main text.

Figure 12: $(a)$ The cubic lattice embedding the active ring. The boundary
points (small red-colored spheres) at the six faces remain fixed, while the
rest of the lattice can distort. The monomers of the chain are bonded with the
lattice points within a cutoff distance $\sigma$. $(b)$ A typical
configuration with distortion of the lattice at $Pe=1.$ The bounding box,
indicated by black lines, is solely for visual representation.

##  Appendix F Distortion of embedding matrix

Fig. 12$(a)$ shows the initial cubic lattice with the active ring embedded in
it. Fig. 12$(b)$ shows a typical steady-state configuration with the
distortion of the lattice points and stretching of the bonds at $Pe=1$.

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