# Vesicle formation induced by thermal fluctuations

Andreu F. Gallen<EMAIL_ADDRESS>Departament Física de la Matèria
Condensada, Universitat de Barcelona, E-08028 Barcelona, Spain  J. Roberto
Romero-Arias  Instituto de Investigaciones en Matemáticas Aplicadas y en
Sistemas, Universidad Nacional Autónoma de México, 01000 Ciudad de México,
Mexico.  Rafael A. Barrio  Instituto de Física, U.N.A.M., 01000, Ap. Postal
101000, México D.F., México  Aurora Hernandez-Machado  Departament Física de
la Matèria Condensada, Universitat de Barcelona, E-08028 Barcelona, Spain

###### Abstract

The process of fission and vesicle formation depends on the geometry of the
membrane that will split. For instance, a flat surface finds it difficult to
form vesicles because of the lack of curved regions where to start the
process. Here we show that vesicle formation can be promoted by temperature,
by using a membrane phase field model with Gaussian curvature. We find a phase
transition between fluctuating and vesiculation phases that depends on
temperature, spontaneous curvature, and the ratio between bending and Gaussian
moduli. We analysed the energy dynamical behaviour of these processes and
found that the main driving ingredient is the Gaussian energy term, although
the curvature energy term usually helps with the process as well. We also
found that the chemical potential can be used to investigate the temperature
of the system. Finally we address how temperature changes the condition for
spontaneous vesiculation for all geometries, making it happen in a wider range
of values of the Gaussian modulus.

###### keywords:

American Chemical Society, LaTeX

Institute of Nanoscience and Nanotechnology (IN2UB), 08028 Barcelona, Spain
IR,NMR,UV

##  1 Introduction

Lipid vesicles, also known as liposomes or simply vesicles, are structures
that are crucial for many biological processes, as transport vesicles 1,
secretory vesicles 2, and endocytosis 3. Many of these biological systems
require to create and destroy vesicles constantly, e.g. the Golgi apparatus 4,
5, the synaptic system 6, 7, or enveloped viruses 8, 9. Even red blood cells
are known to loose membrane when they become old through shedding of
microvesicles and leading to a change of their surface-volume ratio 10. Thus,
there is great interest in gaining insight on the dynamics of vesicle
formation, also referred to as vesiculation. Vesicles are also produced
artificially, either from preformed bilayers or by building them directly from
an organic mixture of lipids in contact with an aqueous medium 11, using some
electroformation 12 or thin-film hydration methods13.

In a previous article 14, the vesiculation of a membrane tube was studied by
introducing the Gaussian curvature energy term. This Gaussian term is
necessary to study topological transitions on membranes and has usually been
neglected 15. The membrane tube geometry helped the fission process and leads
to the membrane splitting into multiple vesicles 14. However, there are some
membrane geometries that instead of helping vesicle formation process they
hinder it like, for example, a perfectly flat membrane.

A perfectly flat membrane is a geometry that hinders this budding process as
it lacks any curved region. Without any point more curved than the other, even
if budding is energetically favourable there is no preferred point from where
to start the membrane budding process. This, in turn, stops vesiculation, even
in the case where generating vesicles is energetically favourable. However, in
nature vesiculation happens in wide variety of situations and with plenty of
membrane geometries for the initial membrane. It is known that vesicle
formation is a temperature-dependent process 3, 16 and that temperature
gradients affect the phase behaviour heterogeneous membranes 17 and can change
the lipid phase between fluid and gel states 18. In experimental works, it has
been shown that the size of the vesicles depend on preparation methods, the
type of lipid used and the temperature 19 and even the temperature has been
using to produce fission of vesicles in Giant Unilamelar Vesicules (GUVs)20
considering the first order phase transition between the gel and fluid states
in the lipids.

The processes beyond membrane fission has been challenging. There are some
works in microscopic modelling with molecular dynamic simulations21, 22, 23,
but in the mesoscopic modelling scales little has been done, and there is a
bit understanding on topological transitions 24, 25. In mesoscopic models, one
needs to consider the energy of the membrane as a continuous surface, instead
of going molecule by molecule. Thus, in this work, we study some mechanisms
that promote –or even induce– the formation of a vesicle from a non-favourable
geometry.

To promote the vesiculation from a completely flat membrane, one introduces
the effect of finite temperatures to the previously mentioned model14 and we
develop a three dimensional model to study the dynamics of spontaneous vesicle
formation from a flat membrane by adding a thermal noise. With these
considerations, one shall expect to deform the membrane enough and start the
vesiculation process. However, it is not as simple as that, as there is an
energy barrier that is impeding the flat membrane to change shape and form
buds. Thus, we assemble a phase diagram between fluctuation and vesiculation
states and we find that the topological stability of the membrane depends on
the bending and Gaussian moduli, the spontaneous curvature and the temperature
of the system. Also, we find that temperature is only needed to trigger the
process but that when one has a curved enough region the vesiculation process
will go on, even when turning the temperature back to zero. Surprisingly, we
see no topological transition for positive Gaussian modulus. One would expect
the membrane to form tunnels and channels but the topology of the system
remains unchanged even for excessively large temperatures.

##  2 Model

The model follows the Canham-Helfrich free energy and its phase-field
approaches used for fission of membrane tubes14. The Canham-Helfrich free
energy is written as

$F=\int_{A}\Big{(}\frac{\bar{\kappa}}{2}(C-C_{0})^{2}+\bar{\kappa}_{G}K+\gamma_{A}\Big{)}dA\,+\int_{V}\Delta
p\,dV\text{,}$ (1)

where $\bar{\kappa}$ and $\bar{\kappa}_{G}$ are the bending and Gaussian
moduli, respectively. $C$ is the total curvature, $C_{0}$ is the spontaneous
curvature and $K$ is the Gaussian curvature. The parameters $\gamma_{A}$ and
$\Delta p$ are Lagrange multipliers that ensure the conservation of the area
and the volume respectively. These last can be understood as an effective
surface tension and an osmotic pressure that works against changes of area and
volume, respectively.

Usually, by using the Gauss-Bonnet theorem one can avoid the Gaussian
curvature contribution to the energy, but in this work, we consider the
Gaussian contribution since we are interested in studying the formation of
vesicles. The Gauss-Bonnet theorem states that the integral of the Gaussian
curvature over a surface is proportional to the surface Euler characteristic
$\chi(\Omega)$, which is a topological invariant 26. The Euler characteristic
describes the system by the number of holes $g$ and objects $N$ like
$\chi(\Omega)=2(N-g)$. In general, one can describe a system that increases
the number of objects and generate vesicles with $\bar{\kappa}_{G}<0$ and a
system that increases the number of holes when $\bar{\kappa}_{G}>0$24.

From the free energy of equation (1), one can compute the minimum energy
necessary to form a spherical vesicle from a planar membrane. The energy
required, if we neglect the bilayer thickness effect11, reads

$f=(2\bar{\kappa}+\bar{\kappa}_{G})\frac{A}{R^{2}},$ (2)

were $R$ is the radius of the vesicle, $A$ is its area. To have vesicles
generated spontaneously, we need this energy contribution to be negative.
Thus, one can obtain that for the vesicle formation to be spontaneous, one
requires $\bar{\kappa}_{G}>-2\bar{\kappa}$. Therefore, one should find
spontaneous vesiculation depending on the values of $\bar{\kappa}_{G}$ and
$\bar{\kappa}$. The values of the bending modulus are estimate from elastic
properties of the membranes 24. However, the Gaussian modulus needs an
indirect technique for measurement. An experimental27 and theoretical28 value
of Gaussian modulus is estimated as $\bar{\kappa}_{G}\approx-15K_{B}T$. In a
previous work14, we analysed a phase diagram that arise from vesicle formation
and we found three possible stable states that corresponds to: i) membrane
fission when the negative Gaussian modulus $\bar{\kappa}_{G}$ is much larger
than the bending modulus $\bar{\kappa}$; ii) no fission membrane when
$\bar{\kappa}_{G}$ is negative but not much larger than $\bar{\kappa}$; and
iii) multiple self-connected membrane for positive $\bar{\kappa}_{G}$ in which
many holes are developed in a single continuous membrane.

In the phase diagram $(\bar{\kappa}_{G},\bar{\kappa})$, one finds that for
negative Gaussian modulus there is a transition between spontaneous
vesiculation state and a topologically stable state defined by the line
$\bar{\kappa}_{G}>-2\bar{\kappa}$. As one can compute the energy cost to
generate a vesicle, we can compute the energy cost necessary when one has a
Gaussian modulus that does not comply that condition. Thus, in order to
generate a spherical vesicle of area $A=4\pi R^{2}$ with radius $R$ and
$\bar{\kappa}_{G}<0$, the energy required is
$f=4\pi(2\bar{\kappa}-|\bar{\kappa}_{G}|)$. This energy can be interpreted as
an energy barrier $\Delta E$ that prevents the system from generating
vesicles. Introducing now an spontaneous curvature $C_{0}$ 29, we get that the
energy barrier is $\Delta
E=4\pi(2\bar{\kappa}+\bar{\kappa}_{G})(1-RC_{0})^{2}$. Therefore, depending on
whether the thermal energy, $E_{T}=k_{B}T$, is bigger or smaller than this
energy barrier, one obtains spontaneous vesicle formation or not. This is what
will distinguish between the two phases that we study in this paper, the
fluctuating and the vesiculating one.

Vesiculation is an irreversible process, as it increases the entropy of the
system and we can ensure that for a time long enough the system will generate
vesicles. Therefore, for a system with a temperature $T$ the condition for
spontaneous vesicle formation for the Gaussian modulus will be

$\bar{\kappa}_{G}>-2\bar{\kappa}+k_{B}T.$ (3)

With this formulation, the membrane can undergoes spontaneous fission for new
values of $\bar{\kappa}_{G}$ that could not vesiculate before.

Membrane phase field model

One can write the free energy of the membrane from equation (1) as a function
of an order parameter $\phi$ and its derivatives 30. This order parameter
$\phi$ will define if a given point in space corresponds to: the membrane, the
inner, or the outer fluid of the vesicle. We can obtain the dynamic equation
by doing the functional derivative of both of the free energy contributions
and write the temporal evolution as

$\frac{\partial\phi}{\partial
t}=\nabla^{2}\left(\frac{\delta{F_{C_{0}}}}{\delta\phi}+\frac{\delta{F_{G}}}{\delta\phi}\right),$
(4)

in which the mean and spontaneous curvature $F_{C_{0}}$ contribution reads
like 31

${F_{C_{0}}}=\kappa\int_{\Omega}\Big{(}(\phi-\epsilon
C_{0})(\phi^{2}-1)-\epsilon^{2}\nabla^{2}\phi\Big{)}^{2}dV,$ (5)

where $\kappa=3\bar{\kappa}/4\sqrt{2}\epsilon^{3}$ and $C_{0}=c_{0}/\sqrt{2}$.
Similarly, the Gaussian contribution is14

${F_{G}}=\int\bar{\kappa}_{G}KdS=\bar{\kappa}_{G}\frac{3}{4\sqrt{2}\epsilon}\int\sum_{i<j}\frac{(1-\phi^{2})^{2}}{2}\Big{(}Q_{ii}Q_{jj}-Q_{ij}^{2}\Big{)}dV,$
(6)

where Gaussian modulus $\kappa_{G}=3\bar{\kappa}_{G}/(4\sqrt{2}\epsilon)$ and
$Q_{ij}$ is curvature tensor (see appendix for the exact expression). The
dynamical equation (4) can be written explicitly in terms of the phase field
order parameter $\phi$ as

$\begin{split}\frac{\partial\phi}{\partial
t}&=\kappa\nabla^{2}\Big{(}(3\phi^{2}-1-2\phi\epsilon
C_{0})\Phi-\epsilon^{2}\nabla^{2}\Phi+\gamma_{A}\nabla^{2}\phi\Big{)}\\\
&-\kappa_{G}\nabla^{2}\left(\frac{12\phi}{1-\phi^{2}}T_{1}+\frac{2(3\phi^{2}+1)}{(1-\phi^{2})^{2}}T_{2}\right),\end{split}$
(7)

where $\Phi=(\phi-\epsilon C_{0})(\phi^{2}-1)-\epsilon^{2}\nabla^{2}\phi$. The
terms $T_{1}$ and $T_{2}$, which can be found in the appendix, are computed
from various derivatives of $\phi$ 14, 32.

Numerical Implementation

The equation (7) is highly non-linear with the order parameter and therefore
we will be solving numerically. It has been already corroborated that a simple
Euler method is reliable to solve this system given that the time step $\Delta
t$ is small enough 14. We work with a lattice of distance $\Delta x$ between
the points, which is constant along the system and equal for each dimension
$(x,y,z)$. Thus, we define the initial conditions of the order parameter
$\phi(x,y,z)$ and compute the temporal evolution of the dynamic eq. (7). The
finesse part is computing the contribution

$\frac{12\phi}{1-\phi^{2}}T_{1}+\frac{2(3\phi^{2}+1)}{(1-\phi^{2})^{2}}T_{2},$
(8)

as in the bulk $(1-\phi^{2})\to 0$ and this would go towards infinity. We
avoid this by adding a very small imaginary contribution to $\phi$ and
implementing the integration using the residue theorem over the positive
complex plane for equation (8).

In dynamical equation (7), the bending and the Gaussian contributions can be
differentiated by their energy moduli $\kappa$ and $\kappa_{G}$, respectively.
Thus, the Lagrange multiplier for the volume is not necessary since the
dynamic equation is written conservatively following the diffusion equation32
(also known as model B). On the other hand, the Lagrange multiplier for the
area conservation $\gamma_{A}$, that is necessary when working with finite
closed membranes, will not be computed, because we work with a reservoir for
the surface area and the volume.

Thermal fluctuations

The thermal agitation to the membrane is modelled by white noise. This white
noise represents the displacement generated by the random impacts of molecules
on to the membrane due to their random Brownian motion. Those impacts displace
constantly the membrane from its average position moving different points of
the membrane to different directions and different displacements. Therefore,
we consider that thermal fluctuations are delta-correlated thorough space and
time. This means that when one compares two different points of space, the
fluctuations over a long enough time require to have no correlation between
them, therefore being independent. It is necessary to have a noise that its
intensity independent of time and space. From the Fluctuation Dissipation
theorem, we can obtain the relation between the noise variance and the
temperature via its auto-correlation which, for a noise $\xi$, takes the form

$\langle\xi(\mathbf{x},t)\xi(\mathbf{x}^{\prime},t^{\prime})\rangle=\frac{2k_{B}T}{\kappa}\delta(\mathbf{x}-\mathbf{x}^{\prime})\delta(t-t^{\prime}),$
(9)

where $k_{B}$ is the Boltzmann constant and $T$ the temperature. Its inverse
dependence on the bending modulus $\kappa$ comes from the fact that a higher
rigidity reduces the thermal fluctuations of a membrane. This noise will be
acting on the membrane position through the order parameter $\phi$. The white
noise has a mean $\langle\xi\rangle=0$, a standard deviation $\sigma$ as
Gaussian probability distribution. The variance is the first moment of a
single random variable ${\xi}$ is defined by
$\sigma^{2}\equiv\langle[\xi-\langle\xi\rangle]^{2}\rangle$ and is the measure
of how much the values of $\xi$ deviate from the mean value
$\langle\xi\rangle$ 33. The intensity of the noise can be measured with the
variance $\sigma^{2}$ and from the Fluctuation Dissipation theorem, we know
that for a thermal noise this variance is proportional to the temperature by

$\sigma^{2}=\frac{2k_{B}T}{\kappa}.$ (10)

We can identify in equation (9) that the magnitude of the auto-correlation is
the variance $\sigma^{2}$ which can modulate with a Gaussian distribution.

Now, to add this delta-correlated noise into the system, we consider that the
membrane is obtained for the points in space defined by $\phi=0$ and the
interface relax fast to the hyperbolic tangent profile. Thereby, one can
simply add a noise to the $\phi$ value around the points of the diffuse
interface where $-1<\phi<1$. This would be written like

$\displaystyle\frac{\partial\phi(\mathbf{x},t)}{\partial
t}=\nabla^{2}\Big{(}\frac{\delta F_{C_{0}}}{\delta\phi}+\frac{\delta
F_{G}}{\delta\phi}\Big{)}+\xi(\mathbf{x},t)(1-\phi^{2})/\epsilon^{2},$ (11)

where the noise disturbs only the membrane while most of the bulk remains
undisturbed.

Nonetheless the noise has a Gaussian distribution with a mean zero, the
conservation of volume and area can be violated. Even if the mean of the noise
is zero for short time-scales we can find that the overall contribution of the
noise is non-zero. Thus, one has to be careful when introducing temperature to
a membrane system to ensure that the area and volume are conserved. Thus, to
facilitate fission, we study a flat membrane with a reservoir of surface area
and volume. It is known that individual closed vesicles should not be able to
change area and volume 34. This is more difficult to ensure for individual
vesicles than it would be for the system as a whole. The vesicles size depend
on the spontaneous curvature of the membrane 35 in most cases the resulting
vesicles are small 10 and thus the membrane fluctuations cannot be taken as
small for their length-scale. With their sizes the area and volume of the
vesicles slightly change and fluctuate over time. This could be addressed by
adding complex Lagrange multipliers to the model, however with the model as is
one can study whether vesiculation happen or not, its energy, and the dynamics
that bring a membrane to vesiculate. The proper modelling of the vesicles
after they have scissioned from the mother membrane would add little or
nothing given our goals.

##  3 Results

Negative Gaussian modulus

We focus on simulating flat infinite membranes where, as discussed earlier,
there is an energy barrier that needs to be overcomed to reach fission. With
the white noise on the membrane position, we can mimic the effects of
temperature. Thus, with enough thermal energy, we expect the membrane to
produce fission. However, this depends on the interplay between temperature
and the bending and Gaussian rigidities $\kappa$ and $\kappa_{G}$,
respectively. That means that, if the temperature is not large enough, for a
given $\kappa$ and $\kappa_{G}$, one finds that even though the membrane can
fluctuate, the fluctuations will not be enough to bend the membrane enough so
that a bud can be produced.

In the left side of Figure 1, we show a simulation with no noise at
temperature $T=0$. The membrane remains flat and no deformation occurs no
matter how long the simulation runs. When one adds temperature to the flat
membrane, one could start to see fluctuations that take the shape of something
like the right side of Figure 1. In this case, the thermal fluctuations are
not big enough to produce vesicle formation and one can defined as the
fluctuating regime.

Figure 1: Left: Snapshot of a flat membrane under conditions where fission is
energetically favourable. However due to the starting geometry, we see no
change over time on the membrane shape. The parameters used are $T=0$,
$\epsilon=1$, $\kappa=1$, $\kappa_{G}=-10$ and $C_{0}=-0.5$. Right: Snapshot
of a fluctuating flat membrane under conditions where fission is energetically
favourable and with low temperature. The parameters used are $T=1.33\cdot
10^{-5}(IU)$, $\epsilon=1$, $\kappa=1$, $\kappa_{G}=-2$ and $C_{0}=-0.25$. The
color map of the membranes is present to aid height visualisation. Red color
means the membrane goes down and blue color means the membrane goes up.

With enough temperature, we change into the vesiculation phase, where we can
obtain vesicles from a flat membrane like in Figure 2. If we keep the
simulation running, we can see vesicles keep forming until they fill up all
the simulated space. Although the simulated vesicles after detachment do not
conserve area nor volume perfectly the phenomenology of this system is all
contained by the presented model for this kind of geometry. It has been proven
that the even a membrane with a geometry unfavourable for fission, if it is
combined with a temperature, one can obtain fission and vesicle formation.

Figure 2:  Snapshots of a simulation where the membrane generates vesicles.
From a flat membrane the system generates two vesicles from the oscillations
produced by the temperature. The parameters used $T=3.6\cdot 10^{-5}(IU)$,
$\epsilon=1$, $\kappa=1$, $\kappa_{G}=-2$ and $C_{0}=-0.25$. The color map of
the membranes is present to aid height visualization. Red color means the
membrane goes down and blue color means the membrane goes up.

Vesiculation phase diagram

The only remaining question left is how the phase space for vesicle formation
and fluctuating looks in function of the free parameters. For that, we
analysis the parameters that play an crucial role in whether fission happens
or not. From the eq. (3) one sees that the factor of the energy moduli is the
ratio $\kappa_{G}/\kappa$. Using this, the free parameters are: $T$,
$\kappa_{G}/\kappa$, and $C_{0}$.

In Figure 3, we obtain a phase diagram for a fixed spontaneous curvature
$C_{0}=-0.25$ and $C_{0}=-0.10$. For low temperatures and low ratio
$|\kappa_{G}/\kappa|$, we conserve fluctuation regime. On the contrary, when
any of these quantities increases, we obtain vesicle formation. The
spontaneous curvature value helps the vesiculation process. The bigger $C_{0}$
the bigger the drop in energy between a flat surface (with curvature zero) and
the a spherical vesicle of literally any curvature value, as it will reduce
the bending energy contribution $\kappa(C-C_{0})^{2}.$ We can see when
comparing the two plots in Figure 3 that when decreasing the spontaneous
curvature to $C_{0}=-0.10$, there are areas in the phase diagram that change
phase. It is be more pronounced for lower energy moduli ratios of
$\kappa_{G}/\kappa$. However, one notes that the changes in $C_{0}$ generate
quantitatively equal diagrams. Therefore, one can fix spontaneous curvature
$C_{0}$ and explore only a phase diagram for the temperature $T$ and the ratio
$|\kappa_{G}/\kappa|$.

Figure 3:  Phase diagram of vesicle formation from a flat membrane depending
on temperature $T$ and the energy moduli ratio $\kappa_{G}/\kappa$ for a fixed
spontaneous curvature $C_{0}$. The simulations had a size of $40\times
40\times 44$ and a membrane width of $\epsilon=1$.

In a deep analysis, the results in Figure 3 show that there is an energy
barrier that one has to surpass to obtain vesicle formation. This energy
barrier is being surpassed with the temperature. For any combination of
$C_{0}$ and $|\kappa_{G}/\kappa|$ there is a energy barrier of a given
magnitude, which decreases when increasing either $C_{0}$ or
$|\kappa_{G}/\kappa|$. Thus, for a given energy barrier, as it is shown in the
phase diagrams of the Figure 3, one sees how a minimum temperature is needed
to be able to pass from the fluctuating phase into the vesiculating phase.
Also, it is easy to see that from a given $\kappa_{G}$ on-wards vesiculation
happens for extremely low temperatures.This happens as the value of
$\kappa_{G}$ gets closer and surpasses the condition for spontaneous vesicle
generation $\bar{\kappa}_{G}>-2\bar{\kappa}$. Thus, the region where one sees
a transition between fluctuating and vesiculating lies in the values
$\bar{\kappa}_{G}<-2\bar{\kappa}$, when increasing $T$ and where the Gaussian
modulus is not big enough to fulfil spontaneous vesiculation. Otherwise, we
find that if $k_{B}T$ is lower than the energy barrier, we shall not find
vesiculation. Therefore, a system with a given temperature, the phase
transition between spontaneous vesiculation and stable topology is displaced.
This displacement is proportional to $k_{B}T$ and can be represented by a
displacement of the phase transition as is shown in Figure 4.

Figure 4:  Sketch of how the transition on the $\bar{\kappa}$
$\bar{\kappa}_{G}$ phase space moves with the thermal energy. Here we are
using the original moduli $\bar{\kappa}$ and $\bar{\kappa}_{G}$ which have
units of energy instead of the internal expressions used in the phase field
model. The blue line corresponds to the condition
$\bar{\kappa}_{G}>-2\bar{\kappa}$ which is the case of eq (3) for $T=0$. The
other lines correspond to increasing temperature where $0<T_{1}<T_{2}$.

In Figure 4, we show the phase diagram for $\bar{\kappa}_{G}$ and
$\bar{\kappa}$ modulus, respectively. The distance to the transition like
$\bar{\kappa}_{G}>-2\bar{\kappa}$ can be interpreted as an energy barrier. The
farther away a system is from the line the higher the barrier there is.
Therefore, the higher the temperature is needed to make that point in the
phase space transition to the vesiculation regime.

Estimating the real temperature

For the estimation on the temperature, we will be using the Mean Square
Displacement (MSD) $\langle|h|^{2}\rangle$. To this end, we need to compute
the exact position of the membrane over time. We do this by interpolating the
position $\phi(x,y)=0$ to obtain $h(x,y)$. For a flat membrane is easy as we
only need to go through the direction $z$ until finding the $\phi$ value
closest to zero for each position $(x,y)$. This value $z$ will be our height
$h(x,y)=z(x,y)$. With this information, we can use the following equation 36

$\langle|h|^{2}\rangle=\frac{k_{B}T}{2\pi\kappa}q_{min}^{-2}=\frac{A\,k_{B}\,T}{8\pi^{3}\kappa},$
(12)

where $A=L^{2}$ is the surface area of the system and $L$ its length and
$q_{min}$ is the lower cutoff wave-vector and has a value
$q_{min}=2\pi/\sqrt{A}$.

Using this expressions, we can compare the numerically obtained
$\langle|h|^{2}\rangle$ with the temperature $T$ and the bending modulus
$\kappa$. We can easily know $A$ and $\langle|h|^{2}\rangle$ from a simulation
and we can also fix $\kappa$. Therefore, we can find the thermal energy of the
system, from eq. (12), is

$k_{B}\,T=8\pi^{3}\kappa\frac{\langle|h|^{2}\rangle}{A},$ (13)

and it is independent on the scale that we are simulating. Here, the relevant
value is the ratio between the surface area and the MSD. Moreover, the final
temperature depends strongly on the rigidity $\kappa$. The bending modulus can
take a wide range of values, from $\kappa\approx 10k_{B}T_{a}$ to
$\kappa\approx 100k_{B}T_{a}$ where $T_{a}=298$K. From this expression we can
see that for a membrane with double the bending modulus $\kappa$ we will need
double the temperature to have the same MSD.

In the simulations, we are using internal computational units (IU), thus in
order to connect the approximated temperatures of the simulations with the
real temperatures, lets see the mean square displacement of some simulations.
The results of $\langle|h|^{2}\rangle(\Delta x^{2})$ for the average of
various simulations can be seen in the Table 1. Here, we can see that the mean
square displacement $\langle|h|^{2}\rangle(\Delta x^{2})$ and the internal
temperature $T$ are following eq. (13) as the ratio of increase in $T$ is
followed by the MSD. With these, we can compare the thermal energy with the
MSD. Introducing the MSD for $T=0.9\cdot 10^{-5}$ in eq. (13), one obtains
$k_{B}\,T\approx 0.012\kappa.$

$T$ (IU) | $\langle|h|^{2}\rangle(\Delta x^{2})$
---|---
$0.1\cdot 10^{-5}$ | $0.008$
$0.4\cdot 10^{-5}$ | $0.030$
$0.9\cdot 10^{-5}$ | $0.075$
Table 1: Mean square displacements $\langle|h|^{2}\rangle$ obtained for
different temperatures. The results of $\langle|h|^{2}\rangle$ are taken from
the average of three different simulations at the same temperature.

We note that the final temperature depends on the bending modulus of what we
will be simulating, for soft giant unilamelar made of a given lipid like DOPC,
the bending modulus is $\kappa_{DOPC}=15k_{B}T_{a}$ 37, for vesicles or more
sturdier red blood cells, $\kappa_{RBC}=70k_{B}T_{a}$, and for diseased or
drugged red blood cells it can go even higher to $\kappa_{RBCd}=120k_{B}T_{a}$
38. Thus, the resulting thermal energy for each $\kappa_{DOPC}$ and
$\kappa_{RBC}$ are $k_{B}\,T\approx 0.18k_{B}T_{a}$ and $k_{B}\,T\approx
0.84k_{B}T_{a}$, respectively. Therefore, depending on the bending modulus of
the membrane $\kappa$ at $T_{internal}=0.9\cdot 10^{-5}IU$, we are simulating
around $T=54$K and $T=250$K. This is a wide range of temperatures available to
simulate. The results presented here range mainly from $T_{internal}=0.1\cdot
10^{-5}IU$ up to $T_{internal}=3.6\cdot 10^{-5}IU$, which result in the wide
temperature ranges represented in Table 2. Observe that our results do not
seem to represent adequate temperature ranges for soft membranes, like the
DOPC. Thus our simulations can be used to represent a range of real biological
membranes of high bending rigidity $\kappa$ with realistic temperatures. To
study what happens to softer membranes, one would have to simply simulate
higher temperatures that match the biological range of a given bending modulus
$\kappa$.

Using Table 2, we can plot the region where water is in the liquid state in
the previous phase diagram. In Figure 5, we show the dependency of the bending
modulus in different regions of the phase diagram. With this, we can look for
a given bending if a transition to vesiculation or fluctuation regimes is
possible by changing $T$, $C_{0}$, or $\kappa_{G}$.

|  $T=0.9\cdot 10^{-5}$ IU |  $T=1.6\cdot 10^{-5}$ IU |  $T=3.6\cdot 10^{-5}$ IU
---|---|---|---
$\kappa=15k_{B}T_{a}$ | 54 K | 96 K | 216 K
$\kappa=30k_{B}T_{a}$ | 108 K | 192 K | 432 K
$\kappa=70k_{B}T_{a}$ | 250 K | 444 K | 1000 K
$\kappa=120k_{B}T_{a}$ | 460 K | 850 K | 1850 K
Table 2: Temperatures from $IU$ to Kelvin for different internal values and
different rigidities measured in $k_{B}T_{a}$ where $T_{a}=298$K and $k_{B}$
is the Boltzmann constant. Figure 5:  Phase diagram of vesicle formation
depending on temperature $T$ and the energy moduli ratio $\kappa_{G}/\kappa$
for a fixed spontaneous curvature $C_{0}$. In colours we find different
regions where water is liquid depending on the bending modulus $\kappa$.
Simulations size of $40\times 40\times 44$ and membrane width $\epsilon=1$.

Energy evolution

We study the time evolution of both the Gaussian and the curvature energy
contributions, represented in Figure 6. Here, one can see how for simulations
where there is no vesiculation the energy curves remain flat over time, while
for vesiculating systems the energy contributions decrease. However, looking
at values that both terms take, we can see how the relative change in the
Gaussian contribution is much greater than the one for the curvature term.
Therefore, when a system is at the vesiculating phase, the dynamics of the
system are driven mainly through the decrease in the Gaussian energy term
$F_{G}$ (see eq. (6)). This energy term starts decreasing once the first
vesicles fission from the main membrane, but as there is an energy barrier to
surpass, the simulations that do not have a high enough temperature cannot get
to lower their Gaussian energy term.

Figure 6:  Evolution over time of the two energy contributions for a series of
simulations, in some there is vesiculation and in some there is not. From the
bending free energy we have the Gaussian energy term and the Curvature term
energy density.The curves that remained flat (blue and orange) remain on the
fluctuating phase while the curves that decrease greatly (green and red)
transition to the vesiculation phase.

In Figure 6, we obtain the curvature energy density $F_{C_{0}}/A$ by using eq.
(5) and the surface area of the membrane. This term in general decreases or
remains constant, depending on whether we have vesicle formation or not. In
simulations where we have a normal vesiculation process, the vesicles formed
get a curvature closer to $C_{0}$ than when they where flat ($C=0$) and thus
reduce the term $(C-C_{0})^{2}$ from the free energy. Our interest to study
the energy density of the curvature term instead of the total energy term is
because as the surface area of the membrane increases over time the curvature
energy contribution will also increase over time. This happens because in the
end having membrane has an energy cost in the curvature term of the free
energy, Thus, any increase in surface area increases the curvature term of the
free energy. Meanwhile, the Gaussian energy term due to the Gauss-Bonnet
theorem is proportional to the number of objects and holes. In conclusion the
best way to understand the evolution of the membrane in a vesiculation
scenario where area is not conserved is to plot the complete Gaussian term and
the Curvature term per unit area.

Finally, we study the chemical potential term squared $\int\mu^{2}dV$, where
the chemical potential is defined with the functional derivative $\mu={\delta
F_{C_{0}}}/{\delta\phi}$. This term shows the rate of change in the membrane.
Looking at $\mu^{2}$, in Figure 7, we observe that there is a flat plateau
when the membrane is in the fluctuating state. The value of this plateau is
clearly related to the temperature of the system, as all simulations at the
same temperature overlap their plateaus. In cases where there will be
vesiculation, this term starts climbing up as now there is an additional
contribution to deformation other than temperature.

Figure 7:  Evolution over time of the chemical potential squared or $\mu^{2}$
for different simulations. In the plot there are both simulations where
vesiculation occurs and some simulations where it does not. Curves correspond
to the same simulations as in Figure 6 with the addition of the purple curve.

Different values of $\kappa_{G}$ will make so that the energy contribution of
each vesicle created is bigger with increasing $|\kappa_{G}|$. This changes
the rate of change over time for the Gaussian energy contribution. Thus the
slope of the Gaussian energy term over time will depend on both temperature
$T$ and the Gaussian modulus $\kappa_{G}$. However, in Figure 8, one can see
how by doubling the value of $\kappa_{G}$ from the blue curve to the green one
the slope of the Gaussian energy curve increases drastically, even with a
lower $T$. Thus in this figure, one can see that the rate of change over time
of the energy is very dependent on the Gaussian modulus $\kappa_{G}$. The
temperature can be seen to change also that rate between the two simulations
of Figure 8 that share the same modulus $\kappa_{G}=-1.2$. However,
$\kappa_{G}$ is more influential on the energy terms than the temperature.

Figure 8:  Evolution over time of the two energy contributions for a series of
simulations. From the bending free energy we have the Gaussian energy term and
the Curvature term. In all simulations there is vesiculation and the main
change in the rate at which the energy changes is mediated by the Gaussian
modulus $\kappa_{G}$.

For the curvature energy density in Figure 8, we can see a similar behaviour
as in previous plots for most of the range of $\kappa_{G}$. However, in some
cases the curvature term climbs back again. In these cases the $\kappa_{G}$ is
so big that its dominating over the curvature term and the vesicles deform in
non-spherical ways to try to fit more vesicles at the cost of increasing the
curvature energy term. Also, one can see how after a long time of vesiculation
the Gaussian and Curvature energy terms stop changing and flattens again. This
happens because in these simulations that have been left to run a long time
the system reaches a point where all the space is full of vesicles and there
is no room to put any more.

Positive Gaussian modulus

Changing the sign of Gaussian modulus $\kappa_{G}$ to a positive, one
completely changes the results.Getting the system to transition from the
fluctuating regime seems impossible, even for unrealistically high noise, the
Gaussian contribution can hold the membrane together. The morphology of the
membrane also changes, for very big noises like in Figure 9. We observe huge
deformations and valleys much bigger than the size of the vesicles produced
with negative $\kappa_{G}$. As high as these deformations are and as high as
the temperature, the simulation never shows fission or handles.

Figure 9:  Left: Membrane fluctuating with T=$6.4\cdot 10^{-5}$IU,
$\kappa_{G}=0.4$, and $C_{0}=-0.25$. Right: Membrane fluctuating with
T=$256\cdot 10^{-5}$IU, $\kappa_{G}=0.4$, and $C_{0}=-0.25$. Little matters
the intensity of the noise, for positive Gaussian modulus we find no
topological transitions. For very high noises the membrane curves and stays
curved, but does not go further than that.

Some of the temperatures introduced in Figure 9 are two orders of magnitude
over the previously simulated temperatures. Nonetheless, it seems to not
matter how much thermal energy we introduce the topology of the simulation
never changes. The Gaussian term seems to be holding the membrane together, as
for simulations with $\kappa_{G}=0$ at these extremely high temperatures the
system breaks down. Usually the Gaussian contribution is thought of only
influencing the system when small membrane necks and other shapes close to
topological transitions happen. However, the results obtained seem to suggest
that the Gaussian term is changing radically how the membrane deforms at all
stages and membrane geometries, at least for thermal fluctuations.

The lack of vesiculation makes sense given the Gauss-Bonnet theorem, when
$\kappa_{G}$ is taken positive, predicts that increasing the number of holes
is what would lower the energy of the system. This however, does not happen.
Even if the configuration is hard to achieve, the high temperatures used
should be giving the necessary energy to produce any membrane deformation
necessary. Therefore, maybe the most intriguing of all is not reaching a self-
connected membrane like we see in previous work 14. One would expect with this
level of noise to go through topological transitions but in the direction of
increasing the number of holes and that decreasing the Euler characteristic
and the Gaussian contribution. Thus, it seems that a flat membrane geometry is
even worse for generating membranes with holes and passages than for
generating vesicles.

Turning off the temperature

The temperature is necessary to start the fission process. However, there is
no need to maintain it after a certain time. Even for simulations where there
has still not been a single fission event, if the membrane is already curved
enough due to the temperature, even after turning off the noise it will end in
multiple fission events.

Another consequence of the noise is that the vesicle shape and dynamics are
slightly rough. One can see edges that one would not expect under normal
circumstances as the resulting vesicles sometimes deform into rough shapes.
This can be seen in the left plot of Figure 10. This happens because the
vesicles are rather small and when in comparison with the thermal fluctuations
the latter are big enough to deform the vesicles a lot. As the fluctuations
are rather big at the size scale of a vesicle, it also makes more difficult to
conserve area and volume and, most importantly, to be dominated the bending.
However as stated earlier, the point of these simulations is not to have a
rigorous description of the vesicles after the fission but to study the
process that leads to fission.

Figure 10:  Left:  At extremely high $T$ the membrane shape has more rough
edges and might stick to new vesicles if there is no more room left. Right:
Even after turning the noise off the initially flat membrane keeps generating
vesicles, and the vesicles take their preferred spherical shape. This
simulation was at $T=144\cdot 10^{-5}$ IU, $\kappa_{G}=-0.4$, and
$C_{0}=-0.25$.

To prove that this weird behaviour after fission is due to the noise and not
to problems in the computation or the model, we study what happens if we turn
the noise off mid-way through a simulation. One can see that after an initial
kickoff where the membrane starts to be deformed enough to generate vesicles,
even if turning off the temperature one will still get vesiculation. We can
see this case in Figure 10, where after reaching a point where the membrane
starts to generate vesicles we turn off the temperature and the vesicles
behave more properly and vesiculation continues. This happens because the
fluctuations made the system reach a much better geometry to fission and until
reaching the steady state more vesicles will be formed only due to the
Gaussian energy contribution.

##  4 Discussion

The conditions for spontaneous fission are present, a flat membrane will
fission only in the presence of temperature as one needs a trigger for the
process to begin. Moreover, we see that through temperature membranes that do
not comply the original condition $\bar{\kappa}_{G}>-2\bar{\kappa}$ can
exhibit spontaneous fission. This is of great importance as membranes never
are in a $T=0$ scenario. In the end the thermal energy related to this noise
by $E_{T}=k_{B}T$ is helping the system overcome an energy barrier $\Delta E$
and achieve vesiculation.

There are two possible states in the presence of temperature for the membrane:
either its only fluctuating or its undergoing vesiculation. The phase diagrams
show clearly how the spontaneous curvature $C_{0}$, temperature $T$, and both
Gaussian and bending moduli, $\kappa_{G}$ and $\kappa$, influence in the
transition. While $C_{0}$ changes the position of the transition in the
diagram, it does not change the overall shape of the transition region. This
transition is clearly seen in the region where the condition for spontaneous
vesiculation $\bar{\kappa}_{G}>-2\bar{\kappa}$ is not met and it happens
because the region where the condition is met requires minimal temperature to
trigger the vesiculation and the closer one is to meeting this condition the
lower $T$ is required for fission to start. The phase diagram
$(\kappa_{G},\kappa)$represented this displacement by changing
$\kappa_{G}=-2\kappa+k_{B}T$ (see Figure 4). The results clearly show how
higher spontaneous curvatures help the vesiculation process by lowering the
energy barrier $\Delta E$ to overcome.

For membranes with a positive Gaussian modulus $\kappa_{G}$, there is no
temperature high enough so that it can transition from the fluctuating regime.
We do not observe the membrane use the thermal fluctuations to create neither
vesicles nor handles or passages as we see in 14. One would expect with this
level of noise to maybe go through topological transitions by increasing the
number of holes. This points either to the flat geometry being bad to generate
handles and passages or to the solution being unstable.

All the processes are driven mostly through the Gaussian energy term of the
Helfrich free energy. This is seen in the relative change in the Gaussian
energy term being greater than the energy term related to curvature as seen in
Figure 6. The evolution of the Gaussian energy term seems to scale
exponentially with the Gaussian rigidity $\kappa_{G}$ while the influence of
temperature $T$ is much smaller (Figure 8). Also from a numerical point of
view, there is value on working with the integral of the chemical potential
squared $\int\mu^{2}dx$, which clearly can be used to measure the temperature
of the system.

These results can be exploited to measure the value of the Gaussian rigidity
$\bar{\kappa}_{G}$ on real membranes. By introducing different proteins or
substances that induce an spontaneous curvature $C_{0}$ and then exploring the
whole available range of temperatures one should be able to find the
transition from fluctuating to vesiculation. In the case that no transition is
found, one could introduce a higher $C_{0}$ and explore again all the
available range of temperatures. The temperature at which the transition will
end up happening will depend on the value of $\kappa_{G}/\kappa$. Given that
the value of the bending modulus (or rigidity) $\kappa$ is much easier to
obtain experimentally than $\kappa_{G}$ this could be a method for obtaining
the Gaussian energy modulus.

An open question that remains is whether the different molecular machinery
involved in some fission processes (e.g. the dynamic superfamily 39, 40, 41 or
the ESCRT machinery42, 43) work by lowering the observed energy barrier
$\Delta E$ or simply produce fission regardless of $\Delta E$. There is the
possibility of this depending on the method that each molecular motor involved
in fission uses. All these processes have a chemical rate and this rate is
also dependent on temperature, so experiments on determining the nature on
whether they are dependent on the energetic of the system or not could be
challenging.

A.F.G. acknowledges financial support from MINECO (Spain) project
FIS2016-78883-C2-1-P. Likewise, J.R.R.A. thanks to Ana Pérez-Arteaga and
Ramiro Chávez-Tovar for their technical support, and the IIMAS Department of
Mathematics and Mechanics for allowing him to join its group of researchers.
A.H.M. acknowledges financial support from Ministerio de Ciencia e Innovación
(MICINN, Spain) project PID2019-1060636B-100. J.R.R.A. and R.A.B. are grateful
for the hospitality at the Universitat de Barcelona. R.A.B. was financially
supported by Conacyt through project 283279.

## Appendix

Obtaining the phase field model

To write equation (1) in terms of a phase field order parameter $\phi$ we will
have to first define the total and Gaussian curvatures. Having the surface
normal vector $\hat{n}$ one can define then the curvature tensor

$Q_{ij}=\nabla_{i}\,n_{j},$ (14)

as the normal vector $n$ changes along the surface are directly related to the
curvature of a surface.

Now, using the curvature tensor $Q$ we can write the curvature $C$ and the
Gaussian curvature $K$. The determinant of the curvature tensor $Q$ is always
zero, but using its trace we can write the curvature 32

$C=\text{tr}[Q].$ (15)

The Gaussian curvature is a bit more complex being

$K=\sum_{i,j}\Big{[}\Big{(}Q_{ii}Q_{jj}-Q_{ij}^{2}\Big{)}\frac{1-\delta_{ij}}{2}\Big{]}.$
(16)

The expression is focused on the non-diagonal elements of the tensor because
it is related to the overall shape that the surface takes around a point and
whether is plane, curved, or presents a saddle-splay shape.

To compute the simulations with a simple model that avoid the need of tracking
the interface position we will use a phase field model. Therefore, we define
an order parameter $\phi(\textbf{x})$ that represents if the volume of fluid
at the point x corresponds to either external or internal fluid to the
vesicle. In this article, we choose the values $\phi(\textbf{x})=+1$ for the
internal fluid of the vesicle and $\phi(\textbf{x})=-1$ for the external
fluid. With this order parameter, we can write the curvature parameters as 30

$C[\phi]=\frac{\sqrt{2}}{\epsilon(1-\phi^{2})}\Big{(}-\phi+\phi^{3}-\epsilon^{2}\nabla^{2}\phi\Big{)},$
(17)

and, to compute $K$, we will be using the curvature tensor written for a phase
field

$Q_{ij}=\frac{\sqrt{2}\epsilon}{1-\phi^{2}}\Big{[}\partial_{i}\partial_{j}\phi+\frac{2\phi}{1-\phi^{2}}\partial_{i}\phi\partial_{j}\phi\Big{]},$
(18)

where $\partial_{i}$ refers to a derivative and $i,j$ can be any of our
cartesian coordinates $(x,y,z)$. We will be computing the free energy of eq.
(1) as a phase field which we will split into two different contributions
$F={F_{C_{0}}}+{F_{G}}$, which consists of the Curvature energy term and the
Gaussian curvature term.

The minimisation of this free energy in eq. (1) is computed numerically in
this scheme. In this methodology the main goal is to change from a surface
energy to a volumetric energy by defining an order parameter. For this we use
a surface differential expressed like

$dS=\frac{3}{4\sqrt{2}\epsilon}\Big{(}1-\phi^{2}\Big{)}^{2}dV.$ (19)

Obtain ${\bf T_{1}}$ and ${\bf T_{2}}$

The Gaussian energy term written as a phase field model involve many
derivatives of the order parameter $\phi$. For simplicity, we have called them
$T_{1}$ and $T_{2}$, here we have written explicitly as introduced in 14. The
various derivatives of $\phi$ are written for simplicity as $\phi_{i}$ where
$i$ is either $x$ $y$ $z$

$T_{1}=\phi_{xx}\phi_{yy}+\phi_{xx}\phi_{zz}+\phi_{yy}\phi_{zz}-(\phi_{xy})^{2}-(\phi_{xz})^{2}-(\phi_{yz})^{2},$
(20)

and

$\displaystyle T_{2}=\phi_{xx}(\phi_{y})^{2}+$
$\displaystyle\phi_{xx}(\phi_{z})^{2}+\phi_{yy}(\phi_{x})^{2}+\phi_{yy}(\phi_{z})^{2}+\phi_{zz}(\phi_{x})^{2}+\phi_{zz}(\phi_{y})^{2}$
$\displaystyle-2\phi_{x}\phi_{y}\phi_{xy}-2\phi_{x}\phi_{z}\phi_{xz}-2\phi_{y}\phi_{z}\phi_{yz}.$
(21)

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