# Active buckling of pressurized spherical shells : Monte Carlo Simulation

Vipin Agrawal<EMAIL_ADDRESS>Nordita, KTH Royal Institute of Technology
and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
Department of Physics, Stockholm university, Stockholm,Sweden.  Vikash Pandey
<EMAIL_ADDRESS>Nordita, KTH Royal Institute of Technology and Stockholm
University, Roslagstullsbacken 23, 10691 Stockholm, Sweden  Dhrubaditya Mitra
<EMAIL_ADDRESS>Nordita, KTH Royal Institute of Technology and
Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden

###### Abstract

We study the buckling of pressurized spherical shells by Monte Carlo
simulations in which the detailed balance is explicitly broken – thereby
driving the shell active, out of thermal equilibrium. Such a shell typically
has either higher (active) or lower (sedate) fluctuations compared to one in
thermal equilibrium depending on how the detailed balance is broken. We show
that, for the same set of elastic parameters, a shell that is not buckled in
thermal equilibrium can be buckled if turned active. Similarly a shell that is
buckled in thermal equilibrium can unbuckle if sedated. Based on this result,
we suggest that it is possible to experimentally design microscopic elastic
shells whose buckling can be optically controlled.

Thin spherical shells are commonly found in many natural and engineering
settings. Their sizes can vary over a very large range – from hundred meters,
e.g., the Avicii Arena Stockholm 111 There are several geodesic domes with
sizes ranging from 10 to 200 meters.  down to about hundred nanometers, e.g.,
viral capsules Buenemann and Lenz (2008, 2008); Michel  _et al._ (2006) and
exosomes Pegtel and Gould (2019); Cavallaro  _et al._ (2019). The elastic
properties of shells, including conditions under which buckling can occur,
have been extensively studied Love (2011); Landau and Lifshitz (1970); Koiter
(1963); WT (1976); Pogorelov (1988); Hutchinson (2016). Interest in this
traditional field of applied mathematics has been rekindled in the past
decades because of possible applications to biology and nanoscience
Vliegenthart and Gompper (2006); Gao  _et al._ (2001); Gordon  _et al._
(2004); Buenemann and Lenz (2008); Vella  _et al._ (2012); Vorselen  _et al._
(2017); Wu  _et al._ (2018); Vorselen  _et al._ (2020); Cavallaro  _et al._
(2021); Michel  _et al._ (2006); Zoldesi  _et al._ (2008). For example, the
elastic shell is used as model for nuclear membrane Jackson  _et al._ (2022).
Furthermore, the cell membrane, although often modeled simply as a fluid
membrane, is dynamically tethered to the cytoskeleton – therby acquiring
effective in-plane elastic properties. For example, it has been shown HW  _et
al._ (2002) that to capture the stomatocyte–discocyte–echinocyte sequence of
the human red blood cell within one unified model it is necessary to introduce
nonlinear in-plane shear elastic modulus of the membrane. Numerical
simulations of flowing RBCs that faithfully reproduce experimental
observations also must use nonlinear shear elastic modulus Fedosov  _et al._
(2010a, b); Fedosov (2010); Fedosov  _et al._ (2011); Freund (2014); Zhu  _et
al._ (2014). Crucially, it has been shown that for small enough shells the
thermal fluctuations can bring down the critical buckling pressure by a large
amount Paulose  _et al._ (2012); Košmrlj and Nelson (2017). This opens up the
intriguing possibility of how the elastic properties of shells, in particular
buckling, will change if they are turned active – driven out of thermal
equilibrium.

Figure 1:  Active buckling: Typical snapshots from our simulations for
activity $A=-2$, $0$ and $2$ (from left to right), and pressure $P=0.30P_{\rm
0}$ (top row) and $0.36P_{\rm 0}$ (bottom row) where $P_{\rm 0}$ is the
critical buckling pressure obtained from the mechanical theory of elastic
shells, i.e., at zero temperature. We use $N=5120$, $\mbox{FvK}=4616$, and
$\mbox{ET}=8$. The middle column, $A=0$, corresponds to shells in thermal
equilibrium – an unbuckled shell buckles upon increasing $P/P_{\rm 0}$ from
$0.30$ to $0.36$. This is consistent with the results of Refs. Paulose  _et
al._ (2012); Košmrlj and Nelson (2017). Top row: As activity is increased to
$2$ (right column) the shell buckles. Bottom row: Whereas as activity is
decreased to $-2$ (left column) the shell, that was buckled in thermal
equilibrium, does not buckle at the same pressure.

The fundamental property of living matter is that they are not in thermal
equilibrium Schrödinger (2012) even when they are statistically stationary.
They are active – they consume energy and generate entropy Gnesotto  _et al._
(2018). The statistical and mechanical properties of active matter is a
current topic of considerable interest Marchetti  _et al._ (2013); Ramaswamy
(2010). The fluctuations of the membrane of living cells have active
components, in addition to the thermal fluctuations, due to, e.g., driving by
the active cytoskeleton Peng  _et al._ (2013); Turlier  _et al._ (2016);
Biswas  _et al._ (2017); Turlier and Betz (2019); Manikandan  _et al._ (2022).
Active shells can also be synthetically designed, e.g., by embedding certain
proteins, who acts as active pumps when irradiated with light of certain
frequencey, in bi-lipid membranes Girard  _et al._ (2005); Manneville  _et
al._ (2001). Shells made out of hard–magnetic elastomers can be turned active
by an external magnetic field Yan  _et al._ (2021).

In this paper, we study the buckling of pressurized active spherical shells
using the Monte Carlo (MC) simulations Kumar and Dasgupta (2020) in which
detailed balance is explicitly broken – thereby driving the shell active, out
of thermal equilibrium. Such a shell typically has either higher or lower
fluctuations than thermal ones depending on how the detailed balance is
broken. We call such non-equilibrium stationary states active and sedate
respectively. We show that, within the right range of elastic parameters, a
shell that is not buckled in thermal equilibrium can be buckled if turned
active. Similarly a shell that is buckled in thermal equilibrium can unbuckle
if turned sedate, see Fig. (1). Based on our study, we suggest that it is
possible to experimentally design microscopic elastic shells whose buckling
can be optically controlled.

Let us briefly summarize, following Refs. Paulose  _et al._ (2012); Košmrlj
and Nelson (2017) the model and the key results of theory of thin elastic
shells in thermal equilibrium. A pressurized elastic shell is described by an
effective Hamiltonian, $G_{\rm eff}=G_{\rm 0}+G_{\rm 1}$ where,

$\displaystyle G_{\rm 0}[f]$ $\displaystyle=\frac{1}{2}\int{\rm
d}^{2}{\bm{x}}\left[B(\nabla^{2}f)^{2}-\frac{PR}{2}\mid{\bm{\nabla}}f\mid^{2}+\frac{Y}{R^{2}}f^{2}\right],{~{}\text{and}}$
(1a) $\displaystyle G_{\rm 1}[f]$ $\displaystyle=\frac{Y}{2}\int{\rm
d}^{2}{\bm{x}}\left[\left(\frac{1}{2}\mathcal{P}_{ij}\partial_{i}f\partial_{j}f\right)^{2}-\frac{f}{R}\mathcal{P}_{ij}\partial_{i}f\partial_{j}f\right]\/.$
(1b)

Here ${\bm{x}}=(x_{1},x_{2})$ is a two-dimensional Cartesian coordinate system
and $\mathcal{P}_{ij}\equiv\delta_{ij}-\partial_{i}\partial_{j}/\nabla^{2}$ is
the transverse projection operator. The out-of-plane displacement is
$h(\bm{x})=f_{0}(\bm{x})+f(\bm{x})$ where $f_{0}(\bm{x})$ is the uniform
contraction of the sphere in response to the external pressure. The difference
between the external and the internal pressure is $P$. The part $G_{\rm 0}$ is
harmonic and the part $G_{\rm 1}$ is anharmonic. In this model, we assume the
shell to be amorphous and homogeneous with radius $R$, bending modulus $B$ and
(two–dimensional) Young’s modulus $Y$. Two non-dimensional numbers determine
the elastic behavior of such shells, the Föppl–von-Karman number and the
Elasto-thermal number, defined respectively as

$\mbox{FvK}\equiv\frac{YR^{2}}{B}\/,\quad\mbox{ET}\equiv\frac{k_{\rm
B}T}{B}\sqrt{\mbox{FvK}}\/,$ (2)

where $k_{\rm B}$ is the Boltzmann constant and $T$ is temperature. At
constant ET, the effects of anharmonicity increases with FvK whereas at
constant elastic modulii the effects of thermal fluctuations increases with
ET. Ignoring the anharmonic contribution, using standard tools of equilibrium
statistical mechanics it is straightforward (Paulose  _et al._ , 2012, Eq. 4)
to calculate the spectrum of fluctuations

$S(\bm{q})\equiv\left\langle\hat{f}(\bm{q})\hat{f}(-\bm{q})\right\rangle=\frac{k_{\rm
B}T}{a\left(Bq^{4}-\frac{PRq^{2}}{2}+\frac{Y}{R^{2}}\right)}\/,$ (3)

where $\hat{f}(\bm{q})$ is the Fourier transform of $f({\bm{x}})$ and $a$ is
the area of integration in the $(x_{1},x_{2})$ plane. In equilibrium, the
symbol $\left\langle\cdot\right\rangle$ denotes thermal averaging; whereas for
active cases, it denotes averaging over the non–equilibrium stationary states.
Note that $S(\bm{q})$ blows-up for

$\displaystyle P$ $\displaystyle=P_{\rm
0}\equiv\frac{4B}{R}q_{\ast}^{2}\/,\quad\text{where}$ (4a) $\displaystyle
q_{\ast}$
$\displaystyle\equiv\left(\frac{Y}{BR^{2}}\right)^{1/4}=\frac{\mbox{FvK}^{1/4}}{R}\/,$
(4b)

where $P_{\rm 0}$ is the buckling pressure, independent of temperature,
obtained within the traditional theory Hutchinson (2016) of buckling of
pressurized shells. For a large Föppl–von-Karman number, $q_{\ast}>1/R$ is the
buckling mode. Refs. Paulose  _et al._ (2012); Košmrlj and Nelson (2017) used
renormalization group (RG) techniques to show that the effects of the
anharnomic terms is to renormalize the parameters appearing in the bare
theory, i.e., $P$, $B$, and $Y$ in (3) must be replaced by their
scale–dependent, renormalized versions, see Ref. (Košmrlj and Nelson, 2017,
Eq. 18). Consequently both the pressure and the critical buckling pressure are
renormalized and buckling is obtained if both of these quantities are equal
for a length scale which must be smaller than the radius of the sphere Košmrlj
and Nelson (2017). The results of this RG analysis were validated by Monte
Carlo simulations of spherical shell, randomly triangulated with $N$ grid
points, with discretized bending and stretching energies that translate
directly into a macroscopic elastic modulii Paulose  _et al._ (2012); Gompper
and Kroll (2004); Itzykson (1986). Our Monte Carlo code, described in detail
in Ref. Agrawal  _et al._ (2022), closely follows that of Ref. Paulose  _et
al._ (2012), and faithfully reproduces these results. We incorporate activity
into this model in the following manner.

Over the years, many theoretical models Ramaswamy  _et al._ (2000); Rao and
R.C. (2001); Loubet  _et al._ (2012); Maitra  _et al._ (2014); Hawkins and
Liverpool (2014); Yin  _et al._ (2021); Goriely (2017), have been suggested to
incorporate the effects of active fluctuations into models of membranes. We
use a method that is well suited to use the Monte Carlo setup and has been
used before to study Ising models out of equilibrium Künsch (1984); Godreche
and Bray (2009); Godrèche (2013); Kumar and Dasgupta (2020) – the idea is to
break detailed balance while preserving stationarity. In equilibrium Monte
Carlo simulations two common choices of the transition rate from one state to
another are the Metropolis ($W_{\rm Met}$) and the Glauber ($W_{\rm Gla}$),
given respectively by,

$\displaystyle W_{\rm Met}$ $\displaystyle={\rm
min}\left[1,\exp\left(-\frac{E}{k_{\rm B}T}\right)\right],\/\text{and}$ (5a)
$\displaystyle W_{\rm Gla}$
$\displaystyle=\frac{1}{2}\left[1-\tanh\left(\frac{E}{2k_{\rm
B}T}\right)\right]\/,$ (5b)

where $k_{\rm B}$ is the Boltzmann constant, $T$ is the temperature and $E$ is
the difference in energy between the two states. To drive the membrane out of
equilibrium, following Ref. Kumar and Dasgupta (2020), we replace $E$ by
$E+\Delta E$ where $\Delta E$ is a constant. This guarantees that detailed
balance is broken and the amount by which it is broken is $\Delta E$. If
$\Delta E$ is positive (negative) the probability of acceptance of large
fluctuations is decreased (increased). Thus we define a dimensionless quantity
$A=-\Delta E/(k_{\rm B}T)$ such that simulations with positive $A$, active
simulations, have higher fluctuations than equilibrium ones whereas for
negative $A$, sedate simulations, the fluctuations are less than the
equilibrium ones. For most of the simulations reported here we use the
Metropolis algorithm. In some representative cases, for both equilibrium and
non-equilibrium simulations, we have checked that both the Glauber and
Metropolis algorithm gives the same result.

For lipid vesicles in thermal equilibrium, standard techniques of equilibrium
statistical mechanics Turlier and Betz (2019) and micropipet aspiration
experiments show $\Delta\alpha\propto(k_{\rm B}T/4\pi B)\ln\sigma$ where
$\Delta\alpha$ is the areal strain and $\sigma$ is the surface tension. For
active membranes the same proportionality holds but the constant of
proportionality is different Manneville  _et al._ (2001). This experimental
result was captured by the model in Ref. Prost and Bruinsma (1996) which adds
an additional Ornstein–Uhlenbeck noise to the models of thermal membranes. Our
active Monte Carlo scheme, in planar membranes 222A. Fragkiadoulakis, S.K.
Manikandan, and D. Mitra, unpublished. reproduces the results of, Ref. Prost
and Bruinsma (1996) and also the experimental result of Ref. Manneville  _et
al._ (2001).

In summary, we incorporate the technique of active Monte Carlo Kumar and
Dasgupta (2020) into the Monte Carlo algorithm for spherical shells in thermal
equilibrium Paulose  _et al._ (2012); Gompper and Kroll (2004); Agrawal  _et
al._ (2022) to simulate active shells.

Figure 2: Buckling under pressure: Normalized change in volume as a function
of external pressure for (blue) a shell in thermal equilibrium, (green) active
($A=2$), and (magenta) sedate ($A=-2$) for simulations with $\mbox{ET}=8$ and
$\mbox{FvK}=4616$ and number of grid points $N=5120$. Here $\left\langle
V\right\rangle$ is the ensemble average of volume, and $V_{\rm ref}$ is the
average volume at the smallest pressure difference. The error in $\left\langle
V\right\rangle$ are the shades around the solid lines – they are too small to
be visible. The signature of buckling is the sudden large change in volume.
The critical buckling pressure for the thermal case is consistent with Refs
Paulose  _et al._ (2012); Košmrlj and Nelson (2017).

Figure 3: Phase diagram The phase boundary in (a) the pressure–elasto–thermal
number plane for different activities and (b) in the pressure–activity plane
for different elasto–thermal numbers (gray triangles for $\mbox{ET}=7.99$,
olive squares for $\mbox{ET}=2.12$ and cyan cross for $\mbox{ET}=0.03$). In
(a) the region where the buckled phase is obtained in equilibrium is marked by
blue lines. The region where the buckled phase is obtained for $A=4$ is shaded
in light yellow. The region where the buckled phase is obtained for $A=-4$ is
marked by violet lines. Cases marked with a cross uses the Glauber algorithm.
In (a) the phase boundary obtained by RG calculation Košmrlj and Nelson (2017)
is marked by a black line and the simulation results by Ref. Paulose  _et al._
(2012) are represented by red triangles.

In Fig. (2) we show a typical plot of how the volume, $V$, of the spherical
shell changes as the external pressure is increased from a very small value.
The simulations are done in a constant pressure ensemble, hence volume is a
fluctuating quantity. Henceforth, by volume we mean the average volume
$\left\langle V\right\rangle$. The average volume at the smallest pressure
difference, is the reference volume $V_{\rm ref}$. The error in $\left\langle
V\right\rangle$, shown by the shaded regions in Fig. (2) are the variances –
they are too minute to be visible. First consider the shell under thermal
equilibrium. Buckling shows up as a sharp decrease in volume accompanied by a
typical buckled shape, as shown in Fig. (1). The critical buckling pressure,
$P_{\rm c}$, that we obtain is consistent with the results of Refs. Paulose
_et al._ (2012); Košmrlj and Nelson (2017). We show the results of the
simulations for both the active, $A=2$, and the sedate, $A=-2$ cases. For the
former the critical buckling pressure decreases while for the latter the
critical buckling pressure increases.

Next we decompose the fluctuating height field, $f(\theta,\phi)$ in spherical
harmonics, $\mathcal{Y}_{\ell,m}(\theta,\phi)$:

$\displaystyle f(\theta,\phi)$
$\displaystyle=\sum_{\ell,m}\tilde{f}_{\ell,m}\mathcal{Y}_{\ell}^{m}(\theta,\phi)\/,\quad\text{and
define}$ (6a) $\displaystyle S(\ell)$
$\displaystyle=\frac{4\pi}{(2\ell+1)|\tilde{f}_{00}|^{2}}\sum_{m=-\ell}^{\ell}|\tilde{f}_{\ell,m}|^{2}$
(6b)

In supplimentary material we compare typical plots of $S(\ell)$ for buckled
and unbuckled shells. Buckling is accompanied by appearence of a peak in
$S(\ell)$ at a small $\ell$ value. For the equilibrium case, buckling as a
function of external pressure is an equilibrium phase transition with the
amplitude of the peak of $S(\ell)$ at small $\ell$ as the order parameter
Paulose  _et al._ (2012). But buckling at the fixed $P$ and ET as a function
of activity is not an equilibrium phase transition but can be considered as a
dynamical one. Nevertheless, we can still characterize buckling by appearance
of a peak in $S(\ell)$ for small $\ell$.

To obtain the phase diagram we use thirteen values of elasto–thermal number,
for each of which we use seven values of activity. For a fixed choice of
elasto–thermal number and activity we start our simulations with an initial
condition where the shell is a perfect sphere. Then we choose a fixed value of
external pressure and run our simulations till we reach a stationary state,
which for zero activity is the equilibrium state. Whether the shell is buckled
or not is decided by three checks: (a) significant decrease of volume (b) a
peak at small $\ell$ for $S(\ell)$ (c) visual inspection. If the shell is not
buckled we choose a higher external pressure and start our simulations again
from the same initial condition. The buckling pressure, $P_{\rm c}$ obtained
for a set of parameters is given in supplimentary material. This way we mark
out the phase boundary in the pressure–elasto–thermal number plane for
different activities and in the pressure–activity plane for different
elasto–thermal numbers, see Fig. (3). In Fig. (3a) we also plot the phase
boundary, obtained through a RG calculation in Ref. Košmrlj and Nelson (2017),
which agrees reasonably well with our numerical results for zero activity.
Note that for large enough values of ET and $A$ we reach a part of the phase
diagram where the shell is unstable at zero external pressure and can be made
stable only with positive internal pressure. This part of the phase diagram is
not shown in Fig. (3) although the relevant data are included in supplimentary
material. Note that at small ET for the sedate case it is possible to have the
shell remain unbuckled even for pressure higher $P_{\rm 0}$, i.e., the shell
is stabilized.

Several comments are now in order: One, most of our simulations use $N=5120$.
We have repeated some of our simulations with $N=20252$ and obtained the same
buckling pressure. Two, to obtain the buckling pressure we always start from
the same initial condition and imposed a fixed external pressure. Hence, the
lines of phase separation we show, Fig. (3), are not continuous and will be
improved if the phase diagram is sampled in a finer resolution. Three,
experimentally, it is unclear how to implement the sedate regime, negative
$A$. Nevertheless, synthetic membranes that can be turned active ($A>0$)
optically, has been already realized by embedding certain proteins in a bi-
lipid membranes – proteins that act as active pumps when irradiated with light
of certain frequency Girard  _et al._ (2005); Manneville  _et al._ (2001). In
such cases, only a fraction of points on the shell are active. This can be
incorporated in a straightforward manner in our code and it would be
interesting to see how the critical buckling pressure changes as we change the
fraction of active points. Four, bi–lipid membranes are semi-permeable
Phillips  _et al._ (2012). As the shell buckles the fraction of solute
increases, increasing the partial pressure inside the shell. Experimentally,
this can be avoided by using shell with holes in them. We expect, in such
cases the buckling pressure may change by a small amount. Five, as there are
many different models of active elastic material it behooves us to study the
universallity of our result by performing similar simulations in other models.
This is outside the scope of the present work.

Finally, our simulations point towards the intriguing possibility that within
the right range of elastic parameters, a shell that is not buckled in thermal
equilibrium can be buckled if turned optically active. Based on this, we
suggest that it is possible to experimentally design microscopic elastic
shells whose buckling can be optically controlled. In such devices it may be
possible to drive flows at microscopic scales by buckling and unbuckling of
shells, optically.

###### Acknowledgements.

We thank Apurba Dev, A. Fragkiadoulakis, Sreekanth M. Manikandan, Pinaki
Chaudhuri and Bidisha Sinha for useful discussions. We gratefully acknowledge
the use of the following free software packages: meshzoo Caroli  _et al._
(2010); A simple and fast mesh generator , matplotlib (Hunter, 2007) and VisIt
(Childs  _et al._ , 2012). The simulations were performed on resources
provided by the Swedish National Infrastructure for Computing (SNIC) at PDC
center for high performance computing and the HPC facility supported by the
Technical Division at the Department of Physics, Stockholm University. For the
latter, we gratefully acknowledge generous help from Mikica Kocic. We
acknowledge the financial support of the Swedish Research Council through
grants 638-2013-9243 and 2016-05225.

## Code and data availability

The source code used for the simulations of the study is freely available at
https://github.com/vipinagrawal25/MeMC/releases/tag/v1.1 Agrawal  _et al._
(2022). The simulation setup and the corresponding data are freely available
on Zenodo with DOI: 10.5281/zenodo.6772570. Python scripts are included with
the data to generate all the figures.

##  Appendix A Monte Carlo simulation of elastic shells in thermal equilibrium

Figure 4: An example of triangulated mesh at the node $i$. $\alpha_{\rm
ij},\beta_{\rm ij}$ are the angles opposite to the bond $ij$. Shaded part is
the Voronoi region of triangle T at the nodes $i,j-1,j$ – it lies inside as T
is non-obtuse. The nodes are sorted in counterclockwise direction. The image
is adapted from Ref. Agrawal  _et al._ (2022).

We use a Monte Carlo algorithm following Refs. Paulose  _et al._ (2012);
Gompper and Kroll (2004) to study the elastic properties of shells. A version
of our code, which performs thermal simulation of elastic shells, is already
available as an open source software Agrawal  _et al._ (2022). Here we
describe the algorithm in brief and compare the results for elastic properties
of shells in thermal equilibrium with those obtained in Ref. Paulose  _et al._
(2012). How we adapt the same algorithm to study shells out of equilibrium is
described in the main body of this paper.

###  A.1 Algorithm

We start with $N$ randomly chosen points on a sphere. Then, we run a Monte
Carlo simulation, with a repulsive Lennard-Jones (LJ) potential, of these
points moving on the surface of the sphere. Once the surface Monte Carlo (SMC)
has reached thermal equilibrium, we triangulate the points using the algorithm
described in Ref. Caroli  _et al._ (2010). This is our initial configuration.
The distance between two neighboring nodes $i$ and $j$ is called $\ell^{\rm
0}_{\rm ij}$.

We use Monte Carlo Baumgärtner  _et al._ (2013) simulations to update the
positions of the points (${\bm{X}}_{\rm i}$ at node $i$). For a pressurized
shell, the total energy

$E=E_{\rm stretch}+E_{\rm bend}+PV\/,$ (7)

where the stretching and the bending contributions are, respectively,

$\displaystyle E_{\rm stretch}$
$\displaystyle=\frac{1}{2}{\sum}_{i}\frac{H}{2}{\sum}_{j(i)}\left({\bm{X}}_{\rm
ij}-\ell^{\rm 0}_{\rm ij}\right)^{2}\/,\quad\text{where}$ (8a) $\displaystyle
X_{\rm ij}$ $\displaystyle\equiv\lvert{\bm{X}}_{\rm i}-{\bm{X}}_{\rm
j}\rvert\/\quad\text{and}$ (8b) $\displaystyle E_{\rm bend}$
$\displaystyle=\frac{B}{2}{{\sum}}_{i}{\mathcal{A}}_{i}\left({\bm{L}}_{i}-C\hat{{\bm{n}}}\right)^{2}\/.$
(8c)

Here $P$ is the pressure difference between outside and inside the shell, and
$V$ is the volume. The Young’s modulus of the membrane is given by
$Y=2H/\sqrt{3}$ Itzykson (1986). The bending modulus is $B$, $\hat{{\bm{n}}}$
is the outward normal to the surface, $C$ is its spontaneous curvature, and
${\mathcal{A}}_{i}$ is the area of Voronoi dual cell at the node $i$ Agrawal
_et al._ (2022); Meyer  _et al._ (2003); Hege and Polthier (2003). The
operator

${\bm{L}}_{i}=\frac{1}{{\mathcal{A}}_{i}}{{\sum}}_{j(i)}\frac{1}{2}\left[\cot(\alpha_{\rm
ij})+\cot(\beta_{\rm ij})\right]{\bm{X}}_{\rm ij}\/,$ (9)

is the discrete Laplacian Itzykson (1986); Hege and Polthier (2003); Meyer
_et al._ (2003) at the node $i$. Here $\alpha_{\rm ij},\beta_{\rm ij}$ are the
angles opposite to bond ${\bm{X}}_{\rm ij}$ as shown in Fig. (4). We compute
${\mathcal{A}}_{i}$ as follows Hege and Polthier (2003); Meyer  _et al._
(2003). Consider the triangle T in Fig. (4), defined by the nodes $i,j,j-1$.
If T is non-obtuse, the area of shaded region in Fig. (4) is

${\mathcal{A}}_{j(i)}=\frac{1}{8}\left[X_{\rm ij}^{2}\cot(\alpha_{\rm
ij})+X_{\rm ij-1}^{2}\cot(\beta_{ij-1})\right]\/.$ (10)

If T is an obtuse triangle, the shaded region in Fig. (4) lies outside the
triangle T, then if the angle at the vertex $i$ of T is obtuse
${\mathcal{A}}_{j(i)}=\rm{area}(T)/2$ otherwise
${\mathcal{A}}_{j(i)}=\rm{area}(T)/4$. Here ${\rm
area}(T)=(1/2)\lvert{\bm{X}}_{\rm ij}\times{\bm{X}}_{\rm ij-1}\rvert$. The
area ${\mathcal{A}}_{i}$ is obtained by summing up the contributions from all
the triangles similar to ${\mathcal{A}}_{j(i)}$, in Fig. (4), e.g., the
contribution from the triangle T is the shaded area.

To compute the outward normal to the surface, $\hat{{\bm{n}}}$, in Eq. (8c),
we sort the points about node $i$ in a counterclockwise manner. To sort the
neighbors around any node $i$, we rotate the coordinate system such that, the
$z$ axis passes through the point $i$ along the vector ${\bm{X}}_{\rm i}$. In
this coordinate system we sort the neighbors by their azimuthal angle. Note
that unlike Ref. Gompper and Kroll (2004) we do not incorporate self-
avoidance.

###  A.2 Comparison with Paulose et al. Paulose  _et al._ (2012)

We use the radius of the sphere $R=1$ as our unit of length and $k_{\rm B}T=1$
as our unit of energy.

We decompose the fluctuating height field, $f(\theta,\phi)$ in spherical
harmonics, $\mathcal{Y}^{m}_{\ell}(\theta,\phi)$:

$\displaystyle f(\theta,\phi)$
$\displaystyle=\sum_{\ell,m}\tilde{f}_{\ell,m}\mathcal{Y}^{m}_{\ell}(\theta,\phi)\/,\quad\text{and
define}$ (11a) $\displaystyle S(\ell)$
$\displaystyle\equiv\frac{4\pi}{(2\ell+1)\lvert\tilde{f}_{00}\rvert^{2}}\sum_{m=-\ell}^{\ell}\lvert\tilde{f}_{\ell,m}\rvert^{2}\/.$
(11b)

In Fig. (5) we plot the spectra $S(\ell)$ for number of nodes $N=20252$, the
Föppl–von-Karman number $\mbox{FvK}\equiv YR^{2}/B=4616$ and for three values
of the bending rigidity, $k_{\rm B}T/B=7\times 10^{-4}$ (blue), $0.07$ (red)
and $0.18$ (orange). The continuous line shows the theoretical prediction by
Ref. Paulose  _et al._ (2012) the shaded region is the error obtained by using
BAGGing (B.1) – we find reasonable agreement with the theory. Next, following
Ref. Paulose  _et al._ (2012) we measure the mechanical response of a thermal
shell by deforming it equally at the two poles with two point-like
indentations. This is implemented by two harmonic springs that are attached to
the north and the south poles of the shell. In Fig. (5B) we compare the
force–deformation curve from our simulations with those obtained in Ref.
Paulose  _et al._ (2012). Again we obtain quite reasonable agreement. A
detailed comparison of parameters of our simulations with those of Ref.
Paulose  _et al._ (2012) is given in Table 1.

Figure 5: Comparison with Ref Paulose  _et al._ (2012): (a) The fluctuation spectrum $S(\ell)$ versus $\ell$ for different values of bending modulus $B$. The solid lines are theoretical predictions Paulose  _et al._ (2012). The lines with symbols are from our simulations. The shaded area shows the error, see section (B.1). (b) The force–deformation curve from our simulations compared with Ref. Paulose  _et al._ (2012) (yellow squares). We use $N=20292$ and $\mbox{FvK}=4616$.  Parameters | our simulation | Ref. Paulose  _et al._ (2012)
---|---|---
Number of grid points, $N$ |  $5120$ to $20252$ |  $5530$ to $41816$
Activity, $A$ |  $-4$ to $4$ | $0$
$P/P_{\rm 0}$, |  $-4$ to $1$ |  $0.2$ to $1$
Elasto thermal number, ET |  $1.65\times 10^{-2}$ to $68$ |  $10^{-6}$ to $10^{2}$
Föppl–von Kármán , FvK | $4616$ |  $650$ to $35000$
Total MC steps | $10^{6}-5\times 10^{7}$ | $1.25\times 10^{8}$
Table 1: Comparison between our simulations and those in Ref. Paulose  _et
al._ (2012).

###  A.3 Comparison between Glauber and Metropolis algorithm

We run Monte Carlo simulations with both Metropolis ($W_{\rm Met}$) and the
Glauber ($W_{\rm Gla}$), rates:

$\displaystyle W_{\rm Met}$ $\displaystyle={\rm
min}\left[1,\exp\left(-\frac{E}{k_{\rm B}T}\right)\right]\/\text{and}$ (12a)
$\displaystyle W_{\rm Gla}$
$\displaystyle=\frac{1}{2}\left[1-\tanh\left(\frac{E}{2k_{\rm
B}T}\right)\right]\/.$ (12b)

Here $k_{\rm B}$ is the Boltzmann constant, $T$ is the temperature and $E$ is
the difference in energy between the two states. In Fig. (6), we compare the
probability distribution function (PDF) of the energy (in thermal equilibrium)
of individual nodes obtained by the Glauber and the Metropolis rules. As
expected, the PDF shows the same Boltzmann distribution in both cases.

Figure 6: Probability distribution function of energy at each point of the
surface for Glauber and Metropolis algorithm for N=20252, and total MC steps =
$10^{6}$. Figure 7: Probability distribution function of energies of every
node in the stationary state for activity $A=-4,-2,0,2,4$. The distribution
fits an exponential distribution for the thermal (A=0) case. However for the
active case (e.g. A=4), the distribution does not fit single exponential
curve.

##  Appendix B Monte Carlo simulations out of equilibrium

To drive the membrane out of equilibrium, following Ref. Kumar and Dasgupta
(2020), we replace $E$ in (12) by $E+\Delta E$ where $\Delta E$ is a constant.
This guarantees that detailed balance is broken and the amount by which it is
broken is $\Delta E$. If $\Delta E$ is positive (negative) the probability of
acceptance of large fluctuations is decreased (increased). Thus we define a
dimensionless quantity $A=-\Delta E/(k_{\rm B}T)$ such that simulations with
positive $A$, active simulations, have higher fluctuations than equilibrium
ones whereas for negative $A$, sedate simulations, the fluctuations are less
than the equilibrium ones. A complete list of parameters are shown in caption
of table (2).

In Fig. (7) we plot the probability distribution function (PDF) of energy of
individual nodes. For the equilibrium case the tail of the PDF can be fitted
with $\mathcal{P}(E)\sim\exp-(E/k_{\rm B}T)$, where E is the energy of a node.
But for the out-of-equilibrium cases there is no one exponential that can be
used to fit.

Figure 8: Spectra: The spectra, $S(\ell)$, (11), $\ell\geq 2$ and (a)
$P=0.3P_{\rm 0}$ and (b) $P=0.36P_{\rm 0}$ for thermal (blue) active (green,
$A=2$) and sedate (magenta, $A=-2$). The solid lines are the mean of spectra
and shaded regions are the error. Appearance of a peak at small $\ell$
signifies buckling.

The fluctuation spectrum $S(\ell)$ comparing unbuckled and buckled simulations
are shown in Fig. (8). The unbuckled ones, even when out of equilibrium, shows
$S(\ell)\sim\ell^{-4}$ for a range of scales. The buckled ones show
distinctive peak at small values of $\ell$.

To obtain the phase diagram we use thirteen values of elasto–thermal number,
for each of which we use seven values of activity. For a fixed choice of
elasto–thermal number and activity we start our simulations with an initial
condition where the shell is a perfect sphere. Then we choose a fixed value of
external pressure and run our simulations till we reach a stationary state,
which for zero activity is the equilibrium state. Whether the shell is buckled
or not is decided by three checks: (a) significant decrease of volume (b) a
peak at small $\ell$ for $S(\ell)$ (c) visual inspection. If the shell in not
buckled we choose a higher external pressure and start our simulations again
from the same initial condition. The buckling pressure, $P_{\rm c}$ obtained
for a set of parameters is given in table (2). This way we mark out the phase
boundary in the pressure–elasto–thermal number plane for different activities
and in the pressure–activity plane for different elasto–thermal numbers, see
Fig. 3 in main text. In the Fig. 3(a) of the main text, we also plot the phase
boundary, obtained through a RG calculation in Ref. Košmrlj and Nelson (2017),
which agrees reasonably well with our numerical results for zero activity.
Note that for large enough values of ET and $A$ we reach a part of the phase
diagram where the shell is unstable at zero external pressure and can be made
stable only with positive internal pressure. This part of the phase diagram is
not shown, although the relevant data are included in table (2). Note that at
small ET for the sedate case it is possible to have the shell remain unbuckled
even for pressure higher $P_{\rm 0}$, i.e., the shell is stabilized.

ET A | 4 | 2 | 1 | 0 | -1 | -2 | -4
---|---|---|---|---|---|---|---
$1.65\times 10^{-2}$ | 0.70 | 0.70 | – | 0.70 | 0.70 | 0.80 | 1.10
$3.32\times 10^{-2}$ | 0.60 | 0.60 | – | 0.70 | 0.70 | 0.80 | 1.10
$6.63\times 10^{-2}$ | 0.60 | 0.60 | 0.60 | 0.70 | 0.70 | 0.80 | 1.00
$1.33\times 10^{-1}$ | 0.60 | 0.60 | 0.60 | 0.70 | 0.70 | 0.80 | 1.00
$2.65\times 10^{-1}$ | 0.50 | 0.60 | 0.60 | 0.70 | 0.70 | 0.80 | 1.00
$5.31\times 10^{-1}$ | 0.50 | 0.50 | 0.60 | – | 0.70 | 0.80 | 1.00
$1.06$ | 0.40 | 0.40 | 0.50 | – | 0.60 | 0.80 | 1.00
$2.12$ | 0.40 | 0.40 | 0.47 | 0.50 | 0.60 | 0.70 | 1.00
$4.53$ | 0.30 | 0.30 | – | 0.50 | 0.50 | 0.60 | 0.80
$7.99$ | 0.30 | 0.30 | 0.30 | 0.36 | 0.40 | 0.40 | 0.50
$16.99$ | -2.8 | – | – | 0.30 | 0.40 | 0.40 | –
$33.97$ |  $<$-4 | -2.5 | – | 0.30 | 0.30 | 0.40 | 0.50
$67.94$ |  $<$-4 |  $<$-4 | -0.10 | 0.20 | 0.30 | 0.30 | 0.50
Table 2: Buckling pressure($P_{\rm c}/P_{\rm 0}$): for different values of
elasto–thermal number (ET) and activity ($A$) obtained from the simulation
with FvK=4616, $N=5120$, $R=1$ and $k_{\rm B}T=1$, where $N$ is the number of
grid points, $R$ is radius of the shell, $k_{\rm B}$ is the Boltzmann constant
and $T$ is the temperature. We repeated some of the simulations for $N=20252$
and obtained the same buckling pressure. To reach a stationary state,
typically, we need to run for $10^{6}-5\times 10^{7}$ Monte Carlo steps, where
in one Monte Carlo step, we perform $2N$ Monte Carlo iterations to attempt to
update the position a random point. Underlined values have been reproduced
with Glauber algorithm as well.

###  B.1 Estimation of error

To compute the statistical error in a dataset $\Psi$, we partition it into $M$
smaller datasets $\\{\Psi_{1},\ldots,\Psi_{M}\\}$. Then for each partition we
calculate the statistical quantities, e.g., mean, spectrum, etc. The average
over the partitions is the average value we quote and the error is the
standard deviation calculated across the partitions. We use $M=10$. This
technique is called BAGGing or Bootstrap AGGregation Mehta  _et al._ (2019).

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