1. The document describes a theoretical model for the growth of vesicles through accretion of amphiphilic molecules. 2. The model uses linear non-equilibrium thermodynamics to describe the vesicle system and its dynamics in terms of thermodynamic forces and fluxes. 3. Stability analysis of the model predicts that spherical vesicles are unstable at small surface areas but become stable at larger surface areas due to a balance of molecular incorporation and water transport effects.

1. Thermodynamics and stability of vesicle growth
Richard G. Morris1 and Alan J. McKane2
1 University
2 The
of Warwick, Gibbet Hill Road, Coventry, U. K.
University of Manchester, Manchester, U. K.
richard.morris@warwick.ac.uk
January 6, 2014

2. The self-assembly of amphiphilic aggregates
An amphiphilic molecule is both hydrophilic (water
loving) and lipophilic (fat loving) e.g., C12 H25 SO4 Na.
A wide variety of supra-molecular aggregates can be formed by
adding amphiphilic molecules to water under different conditions.
1.1. AMPHIPHILIC MOLECULES, AGGREGATES, AND VESICLES
13
!!
(a)
(b)
(c)
(d)
(e)
Figure 1.3: Characteristic (fluid phase) aggregates for amphiphile-water solutions
as a function of water concentration. The concentration of water required to form
each aggregate increases from left to right i.e., from (a) to (e).
The concentration of water required to form each aggregate
increases from fluid. This right i.e., from in Fig.to (e). the fluid phase,
described as left to example is illustrated (a) 1.2. Within
different aggregates can be formed by varying the amount of water present. Some

3. Vesicles
1.1. AMPHIPHILIC MOLECULES, AGGREGATES, AND VESICLES
1.
2.
11
Figure 1.1: Diagram of a simple spherical vesicle. Here, amphiphilic molecules
are arranged in typical bilayer fashion, with “tails” pointing inwards. A represenDeformed easilyshown which resistant to lateral tension and shear.
tative amphiphile is but are has an ionic sodium sulphate head group and
a hydrocarbon tail of the form Cn H2n+1 . Sodium dodecyl sulphate, mentioned in
Do not1.1, corresponds to n = 12.
Section easily fuse with each other.
3. Characterised by poor permeability; entities outside of the
Golgi apparatus of eukaryotic cells [11]. Moreover, a selectively permeable bilayer
structure underlying construction of almost all biological membranes [12], and
are not easily transported inside.
typifies the
therefore the study of vesicles is relevant to a very wide range of problems. Of
particular interest here is the role of vesicles in pre-biotic chemistry and the

4. Vesicle dynamics
Vesicles display complex morphology, including the phenomenon of
vesiculation.
Vesiculation resulting from growth due
to accretion has been proposed as an
important mechanism in the formation
of primitive cells.
Dynamical behaviour in the presence of
additional surfactant depends on the
particular chemical.
1. Palmitoyl-oleoyl-phosphatidycholine
(POPC)
2. Oleic acid
J. Käs and E. Sackmann, Biophys. J. 60 (1991)

5. The energy of a bilayer membrane
The spontaneous curvature model (1973) due to W. Helfrich (and
others).
The energy of a membrane is given by
Em =
κ
2
(2H − C0 )2 dA,
where C0 and κ are constants, the spontaneous curvature and
bending rigidity respectively.
(1)

6. Mean curvature
The mean curvature of a surface is given by
H ≡ (C1 + C2 )/2 =
ˆ
· n,
where C1 and C2 are the principle curvatures i.e., the maximum
and minimum normal curvatures.
16
CHAPTER 1. INTRODUCTION
(a) Diagram showing a saddle
geometry. For a point on a
two-dimensional surface, normal radii of curvature R are de-
(b) Graph corresponding to the
geometry shown in (a). Planes coincident with the normal are characterised by the angle of rotation
(2)

7. Existing approaches to modelling vesicles
1. Minimization of the membrane energy subject to external
constraints, typically fixed surface area A, and volume
enclosed V .
K ∼ 10−2 J m−2 (i.e., energy of 108 kb T needed to increase
area by 1% on vesicle of radius 1µm at room temp.).
κ ∼ 10−19 J (i.e., 104 kb T ).
∆p ∼ . . . ?
U. Seifert, R. Lipowski, Z.-C. Ou Yang etc.
2. Vesicle behaviour in a shear flow. So-called “tank-treading”
behaviour etc.
C. Misbah, V. Steinberg etc.

8. Linear nonequilibrium thermodynamics
(de Groot and Mazur)
Single, adiabatically insulated system. Characterised by entropy S
and state variables Ai .
Let ∆S and αi ≡ ∆Ai be the deviations from equilibrium, so that
∆S = −
1 n
fij αi αj .
2 i,j=1
(3)
Taking the time derivative
n
dαj
d∆S
=−
,
fij αi
dt
dt
i,j=1
d∆S
=⇒
=
dt
(4)
n
Ji Xi ,
(5)
i=1
where Ji ≡ dαi /dt and Xi ≡ ∂∆S/∂αi = −
n
j=1 fij αj .

9. Continuum approach
System >> Sub-system (mesoscopic) >> microscopic processes
System is characterised by total quantities (e.g., Stot (t)) and local
quantities (e.g., S(x, t)).
Entropy balance is given by
ρ
ds
=−
dt
· J s + σ,
(6)
where s ≡ S/M and ρ ≡ M/V .
n
=⇒ σ =
Ji Xi .
i=1
(7)

10. Linear constitutive relations
The entropy production σ is described in terms of 2n unknowns
(the Ji and Xi ). However, the entropy is fully characterised by only
n unknowns—the state variables Ai .
=⇒ Introduce n constraints to form a closed system of equations.
Fourier’s law: heat flow ∝ temperature gradient.
Fick’s law: diffusion ∝ concentration gradient.
Thermoelectric effect: applied voltage ∝ temperature
gradient.
Onsager (1931):
n
Ji =
Lij Xj ,
j=1
where Lij = Lji .
(8)

11. A simple vesicle system
4.2. GROWTH DUE TO ACCRETION
57
!
Figure 4.2: Vesicle system schematic. The system is formed from two distinct
phases, dilute water-lipid solution and the lipid bilayer, which are partitioned
into three regions, the exterior, membrane and the interior, labeled I, II and
III, respectively. Thermodynamic variables in regions I and III are taken to be
independent of position; there are no diffusion flows, viscous flows, or chemical
potential gradients. Region II, the membrane, is considered to have reached
equilibrium in the sense that the molecules are arranged in the usual bilayer
configuration (shown in the exploded section): “tails” pointing inwards and long
axis orientated along the surface normal. Changes in the fluid resulting from
transport in and out of the membrane are assumed to be confined to very small
areas surrounding the membrane boundary, these areas are labeled IV and V and
are taken to be quasi-stationary. That is, state variables may vary with position
but on the timescale of changes experienced in regions I, II and III, they are
independent of time. The exterior is taken to behave like a large reservoir while,
by contrast, it is assumed that there is no net exchange of lipids between the
Regions I and III are uniform. Region I is a reservoir.
Regions IV and V are quasi-stationary.
The entropy produced due to the re-alignment of existing lipids
when an external molecule is incorporated into the membrane can
be ignored. The corresponding energy change cannot.

12. Vesicle thermodynamics
The entropy production can be written as
T σtot = ∆p
dEm
dA
dV
−
+γ ,
dt
dt
dt
(9)
where ∆p is the pressure difference between the exterior and
interior, and γ is the surface tension.
Assume that changes in the energy of the membrane are
dominated by the addition of lipids to the surface, and by changes
to the pressure difference across the membrane.
E = E (A, V ) =⇒ T σtot = (∆p)eff
dV
dA
+ γeff
,
dt
dt
(10)
∂Em
∂A
(11)
where
(∆p)eff = ∆p −
∂Em
∂V
, γeff = γ −
A
.
V

13. Recap
Thermodynamic fluxes are given by
1 dV
A dt
and
1 dA
A dt .
Thermodynamic forces are (∆p)eff and γeff .
Linked by constitutive relations (recall Ji =
=⇒ How the system approaches equilibrium.
j
Lij Xj ).
Dynamical variables are A(t) and V (t) =⇒ The problem is now
one of geometry.
Deformations can be quantified by using a perturbative approach
˜=r+ n
ˆ
r
In order for the partial derivatives to be of first order in , the terms
involved in the membrane energy Em must be taken to third order.

14. Perturbation theory
Analytical progress can be made (i.e., partial derivatives can be
calculated) if deformations are restricted to be from a sphere.
Integrals can be written in terms of the Laplacian on the surface of
a sphere which can be exploited by writing the deformation as
∞
l
alm Ylm (θ, φ) .
(θ, φ) = ε
(12)
l=2 m=−l
For example, the surface area is given by
∞
A = 4πR 2 + ε2 R 2
l
1
|alm |2 1 + l (l + 1) + O ε4 . (13)
2
l=2 m=−l
Dynamically, the picture is that Em (R(t), ε(t)).

15. Growth due to accretion
System is being driven away from equilibrium. Replace the
constitutive relation for surface area growth by the following
growth condition
dA
= λA =⇒ A(t) = A(0)e λt .
dt
(14)
Remaining constitutive relation is given by
1 dV
= Lp (∆p)eff + Lγ γeff ,
A dt
(15)
The above constraints are not comeplementary to a description in
terms of dynamical variables R(t) and ε(t).
Change variable R(t) −→ r (t), the radius of a sphere of equivalent
area i.e., A = 4πr 2 .

16. Results: spherical growth
Spherical growth corresponds to ε = 0 i.e., the surface area A and
volume V are no longer independent.
The condition which arises is given by
C0 κ
2γ
λR
= ∆p − 2 (C0 R − 2) +
,
2Lp
R
R
where Lγ = 2Lp /r .
(16)

17. Results: stability of perturbations
Focus on single mode perturbations (θ) = εal Yl0 (θ).
al ε
dε
2πLγ C0 κ
= −ε (r − rc1 ) (r − rc2 )
g(l),
dt
r4
dε
= F (r ) (ε = 0) .
dt
(18)
2Lp
,
Lγ
(19)
20l(l + 1) − 6l 2 (l + 1)2
.
C0 (3l 2 (l + 1)2 − 6l(l + 1) + 8)
(20)
=⇒
With
rc1 =
and
rc2 =
(17)

18. Results: stability of single mode perturbations
F (¯) = − (¯ − ¯c1 ) (¯ + |¯c2 |)
r
r r
r
r
¯
2π Lγ
g(l),
al ¯4
r
(21)
F r
0.006
l 6
l 4
l 2
0.004
0.002
20
0.002
0.004
40
60
80
r

19. Physical interpretation
The behaviour predicted is a result of two competing mechanisms:
Water transport is physically
impeded by the
configuration of lipid
molecules.
Lipids may incorporate
themselves more easily from
the exterior.
At small surface areas, the dominant effect is that lipids block the
transport of water and hence the sphere is unstable.
At larger surface areas, the ease with which new lipids can be
incorporated into the membrane causes perturbations to be stable.

20. Ellipsoidal deformations
0
Consider perturbations corresponding to Y2 (θ) = ε 3 cos2 θ − 1
F r
0.010
Oblate
Prolate
0.005
20
40
60
80
r
0.005
0.010
0
0
Oblate ellipsoid: (θ) = −εY2 (θ), prolate ellipsoid: (θ) = εY2 (θ).

21. Summary and conclusions
1. If sufficient care is taken, the linear irreversible dynamics of a
single vesicle can be written-down in rigorous way.
2. It is possible to find analytical expressions that characterise
the stability of a spherical vesicle, which is being driven from
equilibrium by accretion. The results predict behaviour which
is in line with observations made in the laboratory.
3. More experimental input is needed before validity of this
approach can be fully assesed.
R. G. Morris, D. Fanelli, and A. J. McKane, Phys. Rev. E 80 (2010)
R. G. Morris and A. J. McKane, Phys. Rev. E 82 (2011)

23. Discontinuous systems
(de Groot and Mazur)
II
I
III
Regions I and II are uniform. Region II is said to be
quasi-stationary.
Extensive thermodynamic variables scale with the system size.
Assumed to be sufficiently close to equilibrium that timescales
of processes in each region are of the same order.
Rate of change of variables in the capillary are taken to be
negligible.