Becker–DoŽ ring model of self-reproducing vesicles
Peter V. Coveneya* and Jonathan A. D. Wattisb¤
a Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge, UK CB3 0EL
and Department of T heoretical Physics, Oxford University, 1 Keble Road, Oxford, UK OX1 3NP
b Department of T heoretical Mechanics, University of Nottingham, University Park, Nottingham,
UK NG7 2RD
Important developments have been made recently in the experimental study of self-reproducing supramolecular systems based on
micelles and vesicles (or liposomes) ; the processes are related to possible prebiotic transformations involving the forerunners of
biological cells. Here we construct and study a kinetic model which describes both the formation and self-reproduction of vesicles.
Until now, a detailed mechanistic understanding of vesicle formation has been lacking. Our approach is based on a novel
generalisation of the BeckerÈDoŽring cluster equations which describe the stepwise growth and fragmentation of vesicular structures. The non-linear kinetic model we present is highly complex and involves many microscopic processes ; however, by means of
a systematic contraction of the complete set of kinetic equations to the macroscopic limit, we show that the model correctly
captures the experimentally observed behaviour.
1 Introduction
The phenomenon of self-reproduction is an inherent feature of
living systems. It is also central to the question of the origin of
life itself. Much contemporary research concerned with the
search for the origins of life is devoted to a consideration of
the self-replicating properties of individual molecules, including RNA, DNA and their presumed progenitors.1h3 However,
self-reproduction is a form of autocatalysis and autocatalytic
processes can be realised at higher levels than that of individual molecules.3h5 The present paper is concerned with the
analysis of one example of such supramolecular (or
“ emergent Ï) self-reproduction.
In the past few years, signiÐcant progress has been made in
the experimental study of self-reproducing supramolecular
systems, mainly based on micelles and vesicles (or liposomes) ;
the processes were subsequently connected to possible prebiotic transformations involving the forerunners of biological
cells.6h10 These laboratory systems can be regarded as fulÐlling the conditions for “ minimal life Ï as deÐned by Varela et
al.11 The essential features of self-reproduction in these experiments involve the formation of bounded, cell-like structures
(either micelles or vesicles) at or within the boundaries of
which the synthesis of further such structures is initiated.
Some extensions of these experiments have even involved
attempts at “ core-and-shell Ï reproduction, in which both the
“ cell Ï membrane and RNA within the “ cell Ï are simultaneously
reproduced.12,13
The Ðrst such supramolecular self-reproduction experiments
involved micelles. Immiscible ethyl caprylate was hydrolysed
by aqueous sodium hydroxide : as the concentration of caprylate monomer builds up, the critical micelle concentration is
reached at which point there is a dramatic acceleration in the
rate of hydrolysis. The micelles in the aqueous phase dissolve
large amounts of ethyl caprylate, increasing the reaction rate
which in turn produces more micelles through hydrolysis, and
so on. The reaction terminates when all the ethyl caprylate
has been consumed. A subsequent reduction in the pH of the
solution leads to the production of vesicles. SigniÐcantly, it
was also found that if caprylic anhydride is used in place of
* E-mail : coveney=cambridge.scr.slb.com
¤ E-mail : Jonathan.Wattis=nottingham.ac.uk
ethyl caprylate, vesicles are formed directly at the lower solution pH resulting from its hydrolysis, although the kinetics of
the reaction are similar. Thus, under these conditions the
system is comprised of self-reproducing vesicles rather than
self-reproducing micelles. It is worth remarking in passing that
many vesicle systems are either thought or known to be metastable, and generally require some form of energy input (such
as sonication) for their preparation.14
Two experimental systems which lead to self-reproducing
vesicles have been reported by Walde et al.8 In each case, an
immiscible liquid fatty acid anhydride is hydrolysed by an
aqueous phase : in the Ðrst system, caprylic anhydride reacts
with aqueous sodium hydroxide ; in the second, oleic anhydride is hydrolysed by a pH bu†ered aqueous solution. The
liquids are stirred throughout the reaction. The reason for the
di†erent conditions imposed on the caprylic and oleic anhydride systems stems from the fact that the vesicles that caprylate and oleate anions form are known to be stable under
di†erent conditions of pH. In a typical experiment, one Ðnds
that the concentration of produced surfactant monomer
(caprylate or oleate) builds up gradually until a certain point,
whereupon the hydrolysis reaction rapidly accelerates (see Fig.
1 for a set of experimental results showing this e†ect). This is
due to the existence in both systems of a surfactant critical
aggregation concentration at which the monomers aggregate
into vesicles in appreciable quantities. Once these vesicles are
formed, they solubilise the remaining immiscible anhydride
molecules within the aqueous phase, and hence accelerate the
reaction rate owing to a vastly increased interfacial area of
contact between anhydride and the hydrolysing hydrophilic
species. Thus the reaction becomes autocatalytic overall, since
enhanced production of surfactant monomer leads to
increased concentration of vesicles which further accelerate
hydrolysis.
The critical aggregation concentration (c.a.c.) in the case of
vesicle-forming systems is analogous to the critical micelle
concentration (c.m.c.) for surfactant systems that produce
micellar aggregates. Although both these quantities are normally assumed to be unambiguously deÐned, there is a considerable degree of arbitrariness about them. First of all, it is
important to recognise that they are concepts which are
deÐned only at thermodynamic equilibrium. Focusing on the
vesicle case, it is known that as the equilibrium concentration
J. Chem. Soc., Faraday T rans., 1998, 94(2), 233È246
233
Fig. 1 Graph showing the rise in caprylic acid concentration and
corresponding fall in pH. This graph demonstrates the rapid acceleration of the hydrolysis reaction following an induction time of ca. 15
days [taken from Fig. 6(A) of Walde et al.8].
of monomers is increased within, say, an aqueous solution, a
point is reached where monomers associate into vesiclesÈthis
is the c.a.c. Beyond this point, additional monomers contribute to an increasing concentration of vesicles, while the concentration of free monomers remains essentially constant.
However, the critical aggregation concentration is not the
location of a sharp phase transition ; instead it is the concentration of monomers at which the equilibrium fraction of
monomers within vesicles reaches some arbitrary value,
usually taken to be 0.5. In fact, the equilibrium fraction of
monomers within vesicles is itself somewhat arbitrary, since
there is no clear-cut division between “ vesicles Ï and smaller
clusters that are not regarded as vesicles. The equilibrium fraction of monomers within vesicles varies very rapidly around
the c.a.c., so that to an experimentalist it may appear to look
like a phase transition. The same considerations apply in the
case of the critical micelle concentration. While we have previously discussed the theoretical basis for the c.m.c. phenomenon,15,16 we are not aware of any similar discussion of the
c.a.c. in vesicle-forming Ñuids.
The description of these supramolecular vesicular systems is
a major theoretical challenge ; it is also clearly of considerable
importance not only intrinsically, but for origins-of-life studies
and for the wider relevance and applications of these systems
in pure and applied science. Indeed, Farquhar et al.14 have
recently suggested that, at least for certain (bichained) surfactants, vesicle formation is probably more common than has
generally been thought to be the case. Based on their experimental observations, they suggested a possible breakdown
mechanism for vesicles in qualitative terms, but doubted that
formation occurs simply by reversing those processes. On the
other hand, we have recently put forward a detailed quantitative mathematical model of the kinetics of self-reproducing
micelle experiments ;15 in order to do so, we have had to formulate a description of the processes involved in micelle formation in a more general manner than had previously been
described. The approach is based on a novel application of the
fully non-equilibrium BeckerÈDoŽring equations to micelle formation kinetics and is discussed further in Section 2. In the
present paper, we extend this development by formulating a
new generalisation of the BeckerÈDoŽring model which is
applicable to both vesicle formation and vesicle selfreproduction kinetics. Our model may also be applicable to
certain micellar systems ; however, many micellar systems can
be analysed using simpler models, as we have demonstrated in
234
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
an earlier paper.15 To deal with self-reproducing vesicles
requires a more complicated model from the start. Since, here,
we are particularly interested in comparing our results with
experimental observations from a vesicular system, we shall
refer to vesicles throughout the text.
There are three challenges in the modelling of vesicular selfreproduction which we aim to address : (a) to model vesicle
formation on a microscopic level ; (b) to bridge the gap
between microscopic and macroscopic models in a systematic
manner ; and (c) to analyse the resulting macroscopic model to
show that it correctly captures experimentally observed
behaviour.
Sections 2 and 3 introduce the basic theory on which our
models are based. Section 4 addresses the Ðrst challenge in
providing a microscopic model of the mechanisms of vesicle
formation ; Sections 5È7 are necessary for (b), that is to handle
the contraction from microscopic to macroscopic model. In
these sections we derive a simple set of equations for macroscopic and hence observable quantities, from the microscopic
model. Sections 7 and 8, together with the appendices, demonstrate that the solution of our macroscopic model has the
correct properties.
As stated above, a detailed mechanistic understanding even
of simple vesicle formation has until now been lacking.17,18
We emphasize that our model does not assume the existence
of a c.a.c. ; rather that the phenomenon is a consequence of
our model. The kinetic model we present is nonlinear and
highly complex (far more so than our micellar model15)
although it is still a simpliÐed description of these systems ;
however, we are able to show that the model correctly captures the experimentally observed behaviour.
2 Components of the Becker–DoŽ ring model
The BeckerÈDoŽring equations were originally formulated to
describe the kinetics of non-equilibrium gasÈliquid phase transitions on the basis of the reversible processes of droplet formation and growth. Their main aim was to describe the
number density or concentration of droplets of di†ering sizes,
as a function of time. The variable c (t) is used to denote the
r
concentration of clusters C containing r individual monomers
r
(atoms, molecules, ions, etc.). Such a droplet or cluster can
grow or reduce in size via the addition or subtraction of a
single monomer at a time. In chemical notation, the process
may be written
C ]C HC
(2.1)
r
1
r`1
This stepwise aggregationÈfragmentation process is the central
feature of the BeckerÈDoŽring model. Thus, no clusterÈcluster
interactions are allowed in the model. This assumption is a
good approximation in situations where relatively low cluster
concentrations arise (and/or where the interactions are such
that the micelles are known not to coalesce) and there are
appreciable concentrations of monomers (C ) present.
1
In mathematical terms, the kinetics of this model are governed by the equations
dc
r\J
[J ,
r~1
r
dt
r \ 2, 3, . . .
=
dc
1 \ [J [ ; J
r
1
dt
r/1
J \a c c [b c
(2.2)
r
r r 1
r`1 r`1
which are derived from eqn. (2.1) by applying the law of mass
action. In eqn. (2.2), a represents the forward rate of reaction
r
of (2.1), b
the backward rate, and J denotes the Ñux from
r`1
r
clusters of size r to size r ] 1. Since the monomer unit C is
1
involved in every reaction, there is a special equation for its
concentration c (t). The system of equations (2.2) has the
1
following important properties which constrain the kinetics it
describes.
1 A unique equilibrium solution exists. We denote this by
c6 \ Q c6 r , where Q is a cluster partition function (Q \ 1). It
r
r 1
r
1
is connected to the forward and backward rate coefficients by
a Q \b Q .
r r
r`1 r`1
2 The total number of monomers present, including those
that are free and those sequestered within clusters, is constant.
In other words, the matter density, o \ ;= rc (t), is conr/1 r
served.
3 The quantity V \ ;= c [log(c /Q ) [ 1] decreases
r/1 r
r r
monotonically with time. It thus qualiÐes as a Lyapunov function ; in physical terms, it corresponds to the free energy of the
system.
4 There is an alternative way of writing the inÐnite set of
ordinary di†erential equations : given an arbitrary sequence of
numbers Mg N= , the following identity (“ weak form Ï) holds
r r/1
=
=
; g c5 \ ; [g
[ g [ g ]J
(2.3)
r r
r`1
r
1 r
r/1
r/1
which is equivalent to the original di†erential eqn. (2.2).
2.1 A generalisation of the Becker–DoŽ ring scheme :
multi-component nucleation
The basic BeckerÈDoŽring model can be generalised to cover a
number of physically di†erent systems. Previously, we have
used extensions and generalisations of the model to describe
micelle formation15,19,20 and to provide a general theory of
nucleation including inhibition for chemically reacting
systems.21 In the present paper, we shall construct another
generalisation which can be used to model the formation and
chemical transformations of liposomes. As an introduction to
this, we shall Ðrst formulate a model for multi-component
micellisation in which there are two di†erent monomer species
present (A, B) and mixed clusters can form.
We shall denote a cluster containing r atoms of type A and
s atoms of type B by C . We still only allow a single
r, s
monomer to be added or removed at a time, but now there
are two monomers ; type A, denoted C , and type B,
1, 0
denoted C . Thus there two reactions that we have to
0,1
account for
HC
r`1, s
C ]C HC
(2.4)
r, s
0, 1
r, s`1
In modelling this system mathematically we deÐne two Ñuxes,
one for the material growing by the addition of monomer A
(J ), and a separate Ñux (J@ ) for growth by the addition of
r, s
r, s
monomer B.
The equations governing such a system are then
C
c5
r, s
r, s
]C
1, 0
\J
[ J ] J@
[ J@
r~1, s
r, s
r, s~1
r, s
G
r \ 1, 2, . . .
s \ 1, 2, . . .
\J
[ J [ J@
r \ 2, 3, . . .
r~1, 0
r, 0
r, 0
c5 \ [J ] J@
[ J@
s \ 2, 3, . . .
0, s
0, s
0, s~1
0, s
c5
\ [J [ J@ [ ; J
1, 0
1, 0
1, 0
r, s
r, s
c5
\ [J [ J@ [ ; J@
0, 1
0, 1
0, 1
r, s
r, s
J \a c c [b
c
r, s
r, s r, s 1, 0
r`1, s r`1, s
J@ \ a@ c c [ b@
c
(2.5)
r, s
r, s r, s 0, 1
r, s`1 r, s`1
where ; signiÐes summation of all values of r P 0 and s P 0
r, s
with the exception of r \ 0 \ s.
This set of equations was originally proposed by Carr et
al. ;22 it has similar properties to the former system (2.2), viz :
There is a unique equilibrium solution c6 \ Q c6 r c6 s ,
r, s
r, s 1, 0 0, 1
c5
r, 0
where Q
now satisÐes a Q \ b
Q
and
r, s
r, s r, s
r`1, s r`1, s
a@ Q \ b@
Q
(as well as Q \ 1 \ Q ).
r, s r, s
r, s`1 r, s`1
1, 0
0, 1
There are now two conserved quantities, the density of A
and the density of B, o \ ; rc and o \ ; sc .
A
r, s r, s
B
r, s r, s
A Lyapunov function, or free energy, V \ ;
r, s
c [log(c /Q ) [ 1] exists, which satisÐes V0 \ 0.
r, s
r, s r, s
An alternative way of writing the equations as a set of identities, also known as a “ weak form Ï, is as follows :
; g c5 \ ; [g
[ g [ g ]J
r, s r, s
r`1, s
r, s
1, 0 r, s
r, s
r, s
] ; [g
[ g [ g ]J@
(2.6)
r, s`1
r, s
0, 1 r, s
r, s
(In fact, later on we shall not make use of these identities
directly, but we shall ensure that such a “ weak form Ï still exists
after any approximations we make.)
In Section 4, this system will be generalised further to allow
other chemical processes to occur.
3 A coarse-graining approximation
This section discusses the central approximation technique
that we use throughout the current paper. The technique has
been employed in our micelle formation and self-reproduction
model15 as well as in our generalised nucleation theory
(incorporating inhibition and chemical reactions)21 to reduce
the number of equations we need to consider ; at the same
time it reduces the number of (generally unknown) parameters
contained in the model. It also has the advantage of producing simple kinetic equations for macroscopically observable
quantities, enabling the theory to be compared with experimental data. In the present section we summarise the application of the coarse-graining procedure to the basic
BeckerÈDoŽring equations, before extending it to the multicomponent equations in the next section. A more detailed
account of the method can be found in our previous
papers.15,21
The aim is to deÐne new variables x to replace the individr
ual cluster concentrations c such that x represents an
r
r
average over a certain number of the c variables, thereby
r
reducing the overall number of ordinary di†erential equations
requiring solution. The number j is used to denote the
number of c values we clump together to perform this
r
averagingÈthus we aim to make the following deÐnitions
1 j
x \ ; c
,
(r [ 1) ; x À c
(3.1)
r j
(r~2)j`j`1
1
1
j/1
Here, we have kept c 4 x separate from the rest of the con1
1
centrations since it has a special roüle in the BeckerÈDoŽring
system. In order to maintain the same BeckerÈDoŽring structure in the new system, we will have new Ñuxes L from the
r
group of clusters x to x . Thus the di†erential part of the
r
r`1
system will be
=
dx
1 \ [L [ ; jL
(3.2)
r
1
dt
r/1
Indeed, only a system such as this can possibly conserve
density and satisfy a set of identities similar to the original
eqn. (2.3).
The Ðnal relationship to determine is how the Ñux L
r
depends on x , x , x . On average, j monomers need to be
1 r r`1
added to a cluster in X to convert it into a cluster within
r
X . This corresponds to the chemical reaction X ] jX H
r`1
r
1
X , which can be incorporated into the mathematical model
r`1
via the deÐnitions
dx
r\L
[L ,
r~1
r
dt
L \ a x xj [ b x
(3.3)
r
r r 1
r`1 r`1
where a , b
can be treated as new parameters. But, to be
r r`1
more rigorous, it can be shown that they are related to a ,
r
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
235
b
by eliminating c
, c
, ..., c
from the
r`1
(r~1)j`2 (r~1)j`3
rj~2
equations for J
,J
, ..., J
giving
(r~1)j`1 (r~1)j`2
rj~2
a \a
a
ÉÉÉ a
r
(r~1)j`1 (r~1)j`2
rj
b
\b
b
ÉÉÉ b
(3.4)
r`1
(r~1)j`2 (r~1)j`3
rj`1
From this it is possible to see that the new coefficients a , b
r r`1
are connected to the cluster partition function Q in a similar
r
way to a , b : a Q
\b Q
. This procedure is
r r`1 r (r~1)j`1
r`1 rj`1
discussed in more detail in Section V of ref. 21.
The system (3.2, 3.3) has all the required BeckerÈDoŽring
properties : its equilibrium solution is x6 \ Q
x6 (r~1)j`1 ;
r
(r~1)j`1 1
by deÐning the density to be o \ ;= [(r [ 1)j ] 1]x , the
r/1
r
density is exactly conserved. V \ ;= x [log(x /Q
)[1]
r/1 r
r (r~1)j`1
is a decreasing function for all solutions of the system, and
hence is a Lyapunov function. Finally, the identities
=
=
; g x5 \ ; [g
[ g [ jg ]L
(3.5)
r r
r`1
r
1 r
r/1
r/1
hold for any sequence Mg N.
r
Thus we have reduced the size of the BeckerÈDoŽring system
by a factor of (1/j) whilst maintaining the same essential
mathematical structure. The new system is of course an
approximation to the original one, but it is intended that solutions of the contracted system will give approximations to
those of the full system ; an elementary study of the accuracy
of the course-graining procedure has been carried out by
Wattis and King.23 This has been found to be a successful
method in earlier and simpler, although still highly non-trivial,
situations15,21 than the case of vesicle formation currently
under consideration.
We also have the usual BeckerÈDoŽring type of reversible
aggregation process of stepwise addition of a monomer to a
vesicle
C ]C HC
1, 0
i, n
i`1, n
and the stepwise incorporation of anhydride
(4.2)
]SHC
(4.3)
i, n
i, n`1
which has been previously introduced by Mavelli and Luisi ;24
we shall denote the forward rate by kf , and the backward
i, n
rate by kb
. (Note that this is treated as a reversible reaci, n`1
tion.24)
It is the rate of formation and break-up of this new complex
C which is rate-determining in the fast phase of the reaction.
i, n
The crucial step
C
C ]C
]C
(n P 1)
(4.4)
i, n
i`1, n~1
1, 0
during which one adsorbed molecule of anhydride is converted to an additional surfactant molecule within the cluster
together with a free surfactant monomer, produces a long
induction time before a sudden rapid reaction, as the process
(4.4) occurs much faster than (4.1).
Fig. 2 shows the overall structure of the reaction scheme
(4.1)È(4.4), whilst Fig. 3 displays a small portion of the vesicular part of the reaction scheme in much greater detail,
together with rate constants and the allowed pathways. In
Fig. 2 each di†erent superscript symbol indicates a di†erent
form of equation needed to describe the kinetics of that type
of cluster. Vesicles are comprised overwhelmingly of surfactant monomers, that is the important C are those with
i, n
i A n. For this reason we shall restrict the (i, n) space to i [ n.
4 A Becker-DoŽ ring model for vesicle formation
We return now to the self-reproducing vesicle experiments.8,9
In these, simple hydrolysis of anhydride occurs in the absence
of vesicles ; chemically we write this step as
S ] 2C
(4.1)
1, 0
where S represents caprylic anhydride and C
the caprylate
1, 0
monomer, from which vesicles can be formed ; C represents
i, n
a vesicle of i monomers with n molecules of the anhydride
absorbed. This latter notation was used by Mavelli and
Luisi.24 Eqn. (4.1) is intended to represent the direct hydrolysis of the immiscible anhydride by aqueous acid or alkali,
which occurs mainly but not exclusively near the anhydrideÈ
water interface in the stirred reaction vessel.16 (It should be
noted in passing that eqn. (4.1) asserts that two surfactant
anions are produced, rather than one anion and one acid
moeity, as stated in the model of Mavelli and Luisi.24)
Fig. 2 Diagram showing the reaction scheme proposed as a model
for vesicle formation. Each di†erent superscript symbol indicates a
di†erent form of rate equation needed to describe the kinetics of that
type of cluster.
Fig. 3 Details of the various kinetic processes occurring in the BeckerÈDoŽring model of self-reproducing vesicle formation. Upper case letters
denote micellar and vesicular species, while rate constants are indicated in lower case letters.
236
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
Putting the four rate processes of eqn. (4.1)È(4.4) together
and writing down the di†erential equations for the corresponding concentrations (in lower case letters), we Ðnd
=
=
s5 \ [k s [ ; ; J@
0
i, n
n/0 i/n`2
=
=
=
=
c5
\ 2k s [ J [ ; ; J ] ; ; k c
1, 0
0
1, 0
i, n
i, n i, n
n/0 i/n`1
n/1 i/n`1
c5
\ J [ J [ J@
2, 0
1, 0
2, 0
2, 0
[J ]k
c
[ J@
c5 \ J
i, 0
i~1, 0
i, 0
i~1, 1 i~1, 1
i, 0
(i P 3)
c5
i, n
\J
i~1, n
[J
i, n
] J@
[ J@
i, n~1
i, n
]k
c
[k c
(n P 1, i P n ] 3)
i~1, n`1 i~1, n`1
i, n i, n
c5
\ [J
] J@
[k
c
n`1, n
n`1, n
n`1, n~1
n`1, n n`1, n
(n P 1)
[J
] J@
n`1, n
n`2, n
n`2, n~1
[ J@
[k
c
(n P 1)
n`2, n
n`2, n n`2, n
J \a c c [b
c
i, n
i, n i, n 1, 0
i`1, n i`1, n
J@ \ kf c s [ kb
c
(4.5)
i, n
i, n i, n
i, n`1 i, n`1
Here the variables J represent Ñuxes from clusters comi, n
posed of i caprylate monomers and n caprylic anhydride molecules to those with one extra monomer ; similarly J@
i, n
represents the Ñux to vesicles with one extra anhydride molecule. The forward rate coefficient for the addition of
monomer is denoted a and the backward rate b
. For
i, n
i`1, n
the addition of anhydride, the forward rate is written kf and
i, n
the backward one as kb
. Caprylic anhydride is irreversibly
i, n`1
converted to caprylate monomer at the rate k . Finally, and
0
crucially for the overall kinetic behaviour, there is an alternative conversion mechanism, namely that a C vesicle is coni, n
verted to a C
vesicle and a monomer (C ) ; this
i`1, n~1
1, 0
occurs at a rate k and is assumed to be an irreversible
i, n
process.
The system of eqn. (4.5) is a new generalisation of the
BeckerÈDoŽring equations. Thus we must check that the
BeckerÈDoŽring structure is preserved. Indeed, we Ðnd the following properties.
1 There is an equilibrium state : s6 \ 0, c6 \ 0, #n P 1,
i, n
c6 \ Q c6 i . The cluster partition function Q is related to
i, 0
i 1, 0
i
the equilibrium chemical potential of cluster C , and also to
i, 0
the rate constants a , b
via a Q \ b
Q . The
i, 0 i`1, 0
i, 0 i
i`1, 0 i`1
other a s and b s need satisfy no special relationship.
i, n
i, n
2 Density : since the anhydride molecule has approximately twice the mass of a single caprylate monomer, the
quantity
c5
n`2, n
\J
=
=
o \ 2s(t) ] ; ; (i ] 2n)c (t),
(4.6)
i, n
n/0 i/n`1
should be conserved. (We note that, in mixed-micelle and
mixed-vesicle systems comprising two distinct monomeric
forms which cannot mutate from one form to the other, there
will be two conserved quantities, one corresponding to the
conservation of each species.)
3 The quantity V \ s ] ;= ;=
nc
satisÐes the
n/1 i/n`1 i, n
conditions for a Lyapunov function. Since there are trajectories over which V0 \ 0, this is not a strong Lyapunov function. Moreover, it does not correspond to a free energy of the
system (note the di†erent form this V has from the previously
quoted Lyapunov functions).
4
There is a weak form
=
=
g s5 ] ; ; g c5
0,1
i, n i, n
n/0 i/n`1
=
=
[ g )k s ] ; ; ( g
[ g [ g )J
0,1 0
i`1, n
i, n
1, 0 i, n
n/0 i/n`1
=
=
] ; ; (g
[ g [ g )J@
i, n`1
i, n
0,1 i, n
n/0 i/n`2
= =
[ g ] g )k c
(4.7)
] ; ; (g
i`1, n~1
i, n
1, 0 i, n i, n
n/1 i/n`1
The complexity of the model (4.5) is obvious. Hence in Section
5 we shall investigate the possibility of reducing the number of
di†erential equations involved by invoking a coarse-graining
technique. Before doing that, we note some of the properties
of the equilibrium solution of eqn. (4.5).
\ (2g
1, 0
4.1 Equilibrium properties–vesicle formation without
self-replication
From point one above, we see that at equilibrium, there is no
anhydride present in the system at all, either in its original
form or incorporated into vesicles. All the vesicles are of the
“ pure Ï form with concentrations denoted by c (the conceni, 0
trations c with n P 1 are all zero). There is simply a disi, n
tribution of “ pure Ï vesicles across a range of sizes ; we denote
by i* the aggregation number where this distribution peaks.
The cluster partition function, Q can be related to the
i
chemical potential of the vesicles in a similar way to that
which occurs in the treatment of micelles.15 We write the temperature as T , BoltzmannÏs constant as k and the chemical
potential of a cluster of size i in its standard state as kE ; then
i
the chemical potential of a vesicle of size i is
k \ kE ] kT log c
(4.8)
i
i
i, 0
If we arbitrarily set the standard chemical potential of monomers to zero (kE \ 0), then the condition of thermodynamic
1
equilibrium k \ ik implies that
i
1
kE \ [kT log Q
(4.9)
i
i
establishing a relationship between the cluster partition function (Q ) and the chemical potential of a cluster C
in its
i
i, 0
standard state.
Since vesicular clusters comprising only a small number of
monomers are highly unstable, they have associated with
them a large chemical potential, implying in turn small values
for Q and hence small concentrations at equilibrium. Small
i
vesicles tend to either break up into individual monomers, or
incorporate monomers to form larger, more stable, vesicles. In
fact, for geometrical reasons it is unlikely that very small
vesicular clusters can form at all ; such transient clusters are
more likely to exist as lamellar bilayers, with high end energies
due to the exposed hydrophobic tails of the individual surfactant molecules. Hence we expect the chemical potential of
monomer and larger vesicles to be lower than for small
vesicles. This implies the existence of an equilibrium cluster
distribution function which shows the presence of some monomers, a few dimers, fewer trimers, then stays very small for a
considerable range of aggregation numbers, before rising to a
maximum at the most probable vesicle size i*, and Ðnally
reducing again (vesicles of arbitrarily large size are not found).
The most important characteristics of this distribution are
that there is a considerable number of monomers present, a
“ rareÐed Ï region where hardly any vesicular clusters of intermediate size are found, and a region containing a well deÐned
distribution of larger vesicles present in signiÐcant concentrations. Such a size distribution is qualitatively similar to that
encountered in spherical micelle systems, and is a crucial
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
237
Fig. 4 Cluster size against aggregation number showing a reasonably strongly peaked distribution of vesicle sizes [taken from Fig. 5(B)
of Walde et al.8]
feature which enables the coarse-grained contraction procedure of Section 5 to be applied successfully. While in micellar systems one typically Ðnds that the distribution of larger
clusters is sharply peaked about one cluster size, that is the
micelles are quite highly monodisperse, in the case of vesicles
the distribution function is normally broader, owing to the
fact that vesicle solutions are usually more polydisperse. See
Fig. 4 for an example of such a size distribution curve.
Before closing this section, there are two further points we
wish to make. The Ðrst concerns the metastable properties of
many vesicular systems referred to in Section 1. In the present
paper, the thermodynamic stability of vesicles is controlled by
the numerical values of their associated chemical potentials
compared with that of monomers. The model on which our
analysis is based, eqn. (4.5), does not include the possibility
that such vesicles can transform into micellar structures
(spheres, rods or disks) or lamellae, although it is perfectly
possible to include such transformations as well, at the
expense of greatly complicating the analysis.25 It is nevertheless intriguing to point out that some elements of the kinetic
metastability of vesicles are captured by the current model (see
the discussion in Section 8).
The second point concerns the validity of the BeckerÈ
DoŽring scheme for modelling vesicle formation. As noted in
Section 2, the approximation is good when the concentrations
of clusters are low and the monomer concentration is high,
and we shall assume these conditions hold here. Whereas the
one-step nature of the model has wide validity for aqueous
spherical micelles comprised of ionic surfactants (for which the
coulomb repulsion between polar head groups is large), the
domain of validity of such an approximation for vesicular
systems is more restricted. Under more general
conditions, the kinetic equations should then be based on
the so-called coagulationÈfragmentation equations of von
Smoluchowski,26 which admit much more general aggregation
and coalescence processes, once again at the cost of increased
analytical complexity.25 Indeed, we note that some of the
more remarkable observed features of self-reproducing vesicular systems, which involve budding and “ birthing Ï,9 can only
be described in these more general terms.
5 Contraction procedure
In this section we introduce a simpliÐcation that enables the
vast array of eqn. (4.5) to be reduced to a manageable number.
238
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
The procedure makes a combination of assumptions which
are quite reasonable in the context of vesicle formation, and
these we shall now describe.
The Ðrst is that the various processes modelled take place
over widely di†ering timescales. In particular, there is the very
slow dissolution of caprylic anhydride to form caprylate
monomer ; another slow process is the self-assembly of monomers into vesicles. Faster processes include the equilibration
of the vesicle distribution at larger cluster sizes, which is controlled by the rapid exchange of monomers between vesicles
and the surrounding solution.
The reason for the slow rate at which monomers aggregate
into vesicles is that the intermediate stages of very small vesicles are highly unstable. They have a great propensity to
break up (dissociate) and return their constituents to free
monomeric form ; thus their typical lifetime is short. It is for
these reasons that the equilibrium distribution has very few
vesicles of small size.
Over shorter periods of time a vesicle is able to absorb or
expel monomers (exchanging them with the surrounding
solution) and so achieve equilibrium with its neighbouring
vesicle sizes. This short-time relaxation we refer to as “ selfequilibration Ï or “ local equilibration Ï since it refers only to
part of the system reaching a (quasi)equilibrium state. The
system is still evolving on longer timescales, so this is not a
true global equilibrium state.
As we mentioned above, larger vesicles are more stable than
very small ones ; hence there is a natural separation in aggregation number between vesicles and free monomers. Thus we
intend to focus on just two regions, one determining the concentrations of very small clusters, and a second covering the
range of sizes in which vesicles are common. Since experimentalists are not often primarily concerned with the details of the
polydispersity of the vesicle size distribution, it is reasonable
to have one variable describing the concentration of monomers and another providing an averaged description of the
vesicle population.
In summary, our reduction method is based on a combination of the assumptions of (i) a separation of timescales
and (ii) the existence of certain local equilibria. The separation
of timescales gives the opportunity for a subset of the kinetic
processes in our model to approach a state of local equilibrium. A global equilibrium solution of the full model carries
over to a corresponding equilibrium solution of the reduced
model. Although much of the following analysis assumes local
equilibrium of certain parts of the system, it never assumes the
system is in a global equilibrium state, or even close to such a
state.
5.1 Derivation of contracted kinetic equations
Following the contraction procedures carried out on simpler
systems in previous papers,15,21 we suggest the following
coarse-grained contraction for the kinetics of vesicle formation. The coarse-graining approximation is achieved by deÐning a new grid K , K@ , with K [ K \ j and K@ [ K@ \
i n
i`1
i
i
n`1
n
k ; so that K and K@ mark the position of the largest aggren
i
n
gation numbers in the space of the c s, and j , k
repj, m
i~1 n~1
resent the size of the region averaged over to Ðnd x . Thus
i, n
we think of the corresponding “ averaged Ï concentrations x
i, n
as being deÐned by
Ki
K n@
1
; c
(5.1)
;
j, m
j k
i~1 n~1 j/K i~1 m/K n~1{
The Ñux from x to x
is then denoted by L
and that
i, n
i`1, n
i, n
from x to x
by L @ ; these quantities are strongly nonlini, n
i, n`1
i, n
ear in x and s, respectively. Their e†ective forward and back1
ward rate constants are a , b
, a@ , b@
. We now
i, n i`1, n i, n i, n`1
denote the rate coefficient for hydrolysis of anhydride by i
0
instead of k , and by i when the conversion to surfactant
0
i, n
x
i, n
\
occurs within clusters represented by x :
i, n
=
=
s5 \ [i s [ ; ; k L @
0
n i, n
n/0 i/n`2
=
=
x5
\ 2i s [ L
[ ; ; jL
1, 0
0
1, 0
i i, n
n/0 i/n`1
=
=
] ; ; l i x
i, n i, n i, n
n/0 i/n`1
x5
\L
[L
[L@
2, 0
1, 0
2, 0
2, 0
x5 \ L
[L [L@ ]i
x
iP3
i, 0
i~1, 0
i, 0
i, 0
i~1, 1 i~1, 1
x5 \ L
[L ]L@
[L@
n P 1,
i, n
i~1, n
i, n
i, n~1
i, n
]i
x
[i x
iPn]3
i~1, n`1 i~1, n`1
i, n i, n
x5
\ [L
]L @
[i
x
nP1
n`1, n
n`1, n
n`1, n~1
n`1, n n`1, n
x5
\L
[L
]L@
n`2, n
n`1, n
n`2, n
n`2, n~1
[L@
[i
x
nP1
n`2, n
n`2, n n`2, n
x
L \ a x xji [ b
i`1, n i`1, n
i, n
i, n i, n 1, 0
L @ \ a@ x skn [ b@
x
(5.2)
i, n
i, n i, n
i, n`1 i, n`1
These equations contain arbitrary coarse-graining functions
(j , k ), and extra parameters l which depend on the granui n
i, n
larity. We now examine the BeckerÈDoŽring structure of this
system to check that such a coarse-graining does not destroy
any of the essential properties.
1 General identities : for any sequence of numbers Mh N
i, n
the following identity relating the derivatives s5 , x5 to i , i ,
i, n
0 i, n
L , L @ holds
i, n i, n
=
=
h s5 ] ; ; h x5 \ (2h [ h )i s
1, 0
0, 1 0
0,1
i, n i, n
n/0 i/n`1
=
=
] ; ; (h
[ h [ j h )L
i`1, n
i, n
i 1, 0 i, n
n/0 i/n`1
=
=
] ; ; (h
[ h [ k h )L @
i, n`1
i, n
n 0, 1 i, n
n/0 i/n`2
=
=
] l h [ h )i x
(5.3)
] ; ; (h
i`1, n~1
i, n 1, 0
i, n i, n i, n
n/1 i/n`1
2 Conservation of density : as in previous contractions, a
minor redeÐnition of density is required in our new coordinate
system. Thus we deÐne o \ 2s(t) ] ;= ;=
(K
n/0 i/n`1 i
] 2K@ )x (t). Then the weak identities (5.3) establish conservan i, n
tion of density provided j \ k , in other words we must use a
i
n
square, uniform coarse-graining mesh (j \ k \ j). Conservai
n
tion of o also requires l \ j. Since j \ j #i and K \ 1, we
i, n
i
1
have K \ (i [ 1)j ] 1 for i P 1, and K@ \ jn. Thus the
i
n
density is
G
=
=
o \ 2s ] ; ; [2jn ] ji [ j ] 1]x
(5.4)
i,n
n/0 i/n`1
3 Equilibrium : from the identities s6 \ 0 ; x6 \ 0 for
i, n
n \ 1, 2, . . . , then by setting the Ñuxes L
equal to zero, we
i,0
Q
.
Ðnd x6 \ Q x6 K i where a Q \ b
i, 0 K i
i`1, 0 K i`1
i, 0
K i 1, 0
4 Lyapunov function : the function
for all i, n. Such a uniform mesh was described in our earlier
paper on self-reproducing micelles.15 A general mesh where j
i
varies with i does not allow density to be conserved in the
present case. Nevertheless, such a mesh has previously been
used to analyse nucleating systems in the presence of chemical
reactions.21 Here, however, in order to conserve density, our
mesh must have the same spacing in both i and n directions.
Eqn. (5.2) then represent the kinetics of the chemical reactions
S ] 2X
1, 0
X ] jS H X
i, n
i, n`1
rate coe†. \ i
0
forward-rate coe†. \ a@
i, n
backward-rate coe†. \ b@
i, n`1
(i P n ] 2)
G
G
forward-rate coe†. \ a
i, n
backward-rate coe†. \ b
i`1, n
X ]X
] jX
rate coe†. \ i
i, n
i`1, n~1
1, 0
i, n
(n P 1) (5.6)
X ] jX H X
i, n
1, 0
i`1, n
6 Maximal contraction
In this section we aim to take the reduction procedure
described earlier as far as possible without eliminating any of
the rate-determining processes. Our aim is to obtain a system
of equations that we can analyse theoretically ; thus we want
to eliminate as many of the intermediate stages of reactions as
possible without losing the essential kinetics of the process.
This corresponds to taking j as large as possible while
keeping at least one of the reactions with rate constants i .
i, n
Hence we shall attempt to analyse a system with just x ,
1, 0
x ,x
and x variables, as shown in Fig. 5.
2, 0 3, 0
2,1
The kinetics of this system are determined by the following
equations
x5
\ 2i s [ L
[ jL [ jL
] ji x
1, 0
0
1, 0
1, 0
2, 0
2, 1 2, 1
s5 \ [i s [ jL @
0
2, 0
x5
\L
[L
[L@
L
\ a xj`1 [ b x
2, 0
1, 0
2, 0
2, 0
1, 0
1, 0 1, 0
2, 0 2, 0
x5
\L
]i x
L
\ a x xj [ b x
3, 0
2, 0
2, 1 2, 1
2, 0
2, 0 2, 0 1, 0
3, 0 3, 0
x5
\L@ [i x
L @ \ a@ x sj [ b@ x
2, 1
2, 0
2, 1 2, 1
2, 0
2, 0 2, 0
2, 1 2, 1
(6.1)
This system has only one conserved quantity, the total density
o,
o \ 2s ] x ] (j ] 1)x ] (2j ] 1)x ] (3j ] 1)x
1, 0
2, 0
3, 0
2, 1
(6.2)
so we have to analyse a fourth-order system. This will be done
by the use of matched asymptotic expansions, noting that
i \ v @ 1. We choose this notation to emphasize that i ,
0
0
which is the uncatalysed rate of decay of anhydride to surfactant, is a small parameter. In the remainder of this paper
we simplify our notation as follows : a \ a , a \ a ,
2
2, 0 1
1, 0
=
=
V \ s ] ; ; jnx
i, n
n/1 i/n`1
satisÐes V0 O 0.
Thus the coarse-graining procedure will only work if the
functions j , k and l satisfy certain constraints. To sumi n
i, n
marise, the conditions on eqn. (5.2) being the correct contraction of (4.5) are
l
\ j, j \ j, k \ j
i, n
i
n
(5.5)
Fig. 5 Diagram showing reaction scheme of maximally contracted
model
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
239
b \ b , b \ b , a@ \ a@ , b@ \ b@ , i \ i , x \
2
2, 0
3
3, 0
2, 0
2, 1
2, 1
1
x , x \x , x \x , y\x .
1, 0 2
2, 0 3
3, 0
2, 1
Chemically, the rate eqn. (6.1) corresponds to the reaction
scheme
S ] 2X
1
rate coe†. \ v À i
0
forward-rate coe†. \ a À a
1
1, 0
KX H X
1
2
backward rate coe†. \ b À b
2
2, 0
forward-rate coe†. \ a@ À a@
2, 0
jS ] X H Y
2
backward-rate coe†. \ b@ À b@
2, 1
forward-rate coe†. \ a À a
2
2, 0
jX ] X H X
1
2
3
backward-rate coe†. \ b À b
3
3, 0
Y ] X ] jX
rate coe†. \ i À i
(6.3)
3
1
2, 1
where the variables label species as follows
G
G
G
S \ caprylic anhydride
X \ caprylate monomer
1
X \ vesicles of pure caprylate, size K \ j ] 1 molecules
2
X \ vesicles of pure caprylate, size M \ 2j ] 1 molecules
3
Y \ mixed vesicles comprising K molecules of caprylate
with j adsorbed molecules of caprylic anhydride. (6.4)
The Ðrst reaction represents the slow, irreversible conversion of caprylic anhydride to monomer. The monomer can
then (reversibly) form vesicles as shown by the second step.
The third shows the adsorption of caprylic anhydride onto the
vesicles, forming mixed vesicles (Y ). The last step shows that
the mixed vesicles convert incorporated anhydride both to
free monomer and monomer which forms part of a larger
vesicle. These larger vesicles (X ) can also be formed by the
3
addition of monomer to smaller vesicles (X ), as shown in the
2
penultimate step of (6.3).
In physicochemical terms, the approximation involved in
the contraction procedure pulls out of the full, inÐnite set of
possible chemical transformations (and associated rate
equations) the smallest subset which describes the ratedetermining processes involved in the growth and selfreproduction of vesicles. As in the micellar model,15 on
physical grounds we expect that the slowest timescale on
which vesicle growth occurs is dictated by the passage of
monomer through the “ bottleneck Ï of intermediate aggregation numbers (which are very unstable and are only present
in extremely low concentrations) into the regime of higher
vesicle aggregation numbers. On both sides of this bottleneck,
matter can be assumed to equilibrate rapidly.
6.1 Comparison with the scheme of Mavelli and Luisi
The approach adopted by Mavelli and Luisi24 in their
attempt to model self-reproducing vesicles relies heavily on
assumptions of thermodynamic equilibrium between species
present in the reaction mixture. Our model di†ers from theirs
in numerous ways. However, the most important distinction
lies at a fundamental level. As in our work on self-reproducing
micelles,15 we begin from a kinetic description which makes
no assumptions of thermodynamic equilibrium : our concern is
to provide a mechanism for how vesicles form and then selfreplicate. Mavelli and Luisi24 prefer to take as given the existence of self-assembled vesicles : they do not attempt to
describe the processes that must be involved in their formation from monomers. Furthermore, they assume that these
structures are essentially in a state of thermodynamic equilibrium with all other reaction components.
240
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
Thus, none of the detailed multi-step processes which are
present in our model (see Sections 4 and 5) appear in their
formulation. In particular, as regards the contracted model we
derived in Section 5, deÐned by eqn. (6.3) above, Mavelli and
Luisi have no analogue to our vesicular growth and fragmentation processes, KX H X , or X ] jX H X .
1
2
2
1
3
7 Further reduction of the rate equations
The fourth-order system of eqn. (6.1), although much simpler
than our original model (4.5), is still too complex to analyse in
detail. Thus, in this Section, we make a further simpliÐcation
to reduce the mathematical system of rate eqn. to a form analysable by the use of phase planes, a common methodology in
nonlinear dynamics which we previously invoked in our
analysis of self-reproducing micelles.15
To achieve this simpliÐcation, we assume that the three
types of vesicle, small and pure (x ), large and pure (x ) and
2
3
mixed (y), are all in local equilibrium with each other. Mathematically speaking, this amounts to assuming a steady-state
distribution of mass between these kinds of vesicles ; that is,
the Ñuxes between these three moieties are in balance such
that
L @ \ iy \ [L
(7.1)
2, 0
2, 0
Mathematically, this approximation is the Ðrst term in an
asymptotic expansion, where each of L
, L@
and iy are
2, 0 2, 0
large, but the combinations L
] iy and L @ [ iy are
2, 0
2, 0
close to zero relative to any one of L
, L @ or iy. Time
2, 0 2, 0
derivatives will enter in higher order terms, but not at leading
order, thus simplifying two of the equations in (6.1) from differential equations to algebraic equations. The quantities x
3
and y are then related to the rest of the system through simple
relations involving x , a situation which can be referred to as
2
x and y being “ slaved Ï to x .27 This subtle aspect of asymp3
2
totic analysis then enables the whole system to be analysed
without recourse to numerical methods. Clearly, this approximation will not always be valid under general nonequilibrium
conditions ; it requires the rate of equilibration among the different types of vesicles to occur on a much faster timescale
than any other process involved in the scheme. In particular,
it assumes that the release of monomer from caprylic anhydride, and the formation of vesicles from monomers, both
occur on longer timescales than self-equilibration among vesicles. In fact, for these surfactant-based self-reproducing
systems, this is a good approximation since micelle and vesicle
aggregation processes are very fast in comparison with the
slow rate of hydrolysis of aqueous anhydride.
Solving the two equations in (7.1), we can write the concentrations y and x in terms of x :
3
2
a
i
a@
a@x sj
2 xj ]
(7.2)
sj
y\ 2 ,
x \x
1
3
2
b
i ] b@ b
i ] b@
3
3
We then use the conservation of density, o \ 2s ] x ] (j
1
] 1)x ] (2j ] 1)x ] (3j ] 1)y, to eliminate x from the
2
3
2
system, leaving just two ordinary di†erential equations for x
1
and s. This substitution amounts to
A
B
b (i ] b@)(o [ x [ 2s)
1
x \ 3
2
D
x \
3
y\
(o [ x [ 2s)[a (i ] b@)xj ] a@isj]
1
2
1
D
(o [ x [ 2s)a@b sj
1
3
D
D \ b (j ] 1)(i ] b@) ] a xj (2j ] 1)(i ] b@)
3
2 1
] a@sj[(3j ] 1)b ] (2j ] 1)i]
(7.3)
3
The substitution s \ o [ 2s means that the initial condi1
tions are s (0) \ 0 \ x (0). Substituting eqn. (7.3) into the
1
1
equations for x5 and s5 from (6.1), we Ðnd that our system is
1
governed by the two ordinary di†erential equations
x5 \ v(o [ s ) [ (j ] 1)a xK
1
1
1 1
b (s [ x )[b (j ] 1)(i ] b@) ] 21~jja@i(o [ s )j]
1 2
1
] 3 1
D
s5 \ v(o [ s ) ]
1
1
21~jb ja@i(s [ x )(o [ s )j
3
1
1
1
D
(7.4)
An analysis of the unperturbed system (that is, when v \ 0)
reveals that the initial condition is a critical (equilibrium)
point. Hence we expect an induction-type behaviour characteristic of a chemical clock, where very little happens at the
start (cf. ref. 15 and 28). One of the eigenvalues at this point
vanishes and the other is negative, conÐrming the existence of
a centre manifold.29 We can use either matched asymptotic
expansions as in ref. 21 or the perturbative centre-manifold
procedure as carried out in ref. 15 to Ðnd approximate solutions.
Region I. A short region during which the precursor S is
converted into monomer. The concentration of monomer
grows linearly (x B vot), and a very small concentration of
1
vesicles is formed (mass in vesicles v \ o [ x [ 2s B vK).
1
Thus the concentration of anhydride falls linearly, s B
1 [o [ vot], and
2
K ([ht)n
vKa oKKK !
1
exp([ht) [ ;
u B vot,
vB
n!
([h)K`1
n/0
(8.3)
C
where h \ B/(D ] D oj).
0
1
Region II. This region is the long induction region ; the
asymptotic equations only balance if time is rescaled to consider time-intervals of O(v~j@K). During this region the variable
v is “ slaved to Ï u, and u changes very slowly. Physically, this
means that the vesicular concentration passively follows the
change in total caprylate concentration, so that vesicles and
monomers stay in equilibrium with each other :
In this section, we summarise approximate solutions to eqn.
(7.4). Although to this point we have performed many approximations and simpliÐcations, albeit in a systematic and controlled manner, the system is still too complex to Ðnd an exact
explicit solution. However, owing to the presence of a small
parameter (v), it is possible to Ðnd a highly accurate approximate solution, using the method of matched asymptotic
expansions and a perturbative centre-manifold procedure.
8.1 Solution by the method of matched asymptotic expansions
In this method, the temporal evolution of the system is split
into a sequence of regions (four in this case). In each region,
some of the terms in these equations are insigniÐcant, and so
can be neglected, giving solvable equations for that time interval. These solutions are then “ matched Ï to ensure that the
system evolves smoothly from one region to the next, and the
system of equations is thus fully solved, as we describe in
outline here. The details of this process are provided in
Appendix B.
To simplify the procedure, we use the variables u \ s and
1
v \ s [ x . This has the advantage that now there is only
1
1
one O(v) term in the di†erential equation system. These variables are not simply mathematically convenient, they represent macroscopic quantities of interest. The total vesicular
mass [Kx ] (2j ] 1)x ] (3j ] 1)y] is represented by v and
2
3
u denotes the total surfactant mass (v ] x \ o [ 2s). Our dif1
ferential eqn. (7.4) become
Kv(o [ u)j
D ] D (o [ u)j ] D (u [ v)j
0
1
2
Bv
v5 \ Ka (u [ v)K [
1
D ] D (o [ u)j ] D (u [ v)j
0
1
2
where the new constants are deÐned by
B
CA
A
B
(8.2)
(8.4)
C
A
D
B
B
1@j
[jKa Koj(t [ t)]
1
2c
Ka (D ] D oj)B1@j
1 0
1
v(t) B
(8.6)
[jKa Koj(t [ t)]K@j
1
2c
These solutions diverge, and another approximation needs to
be found for the concentrations as they approach their equilibrium values. This behaviour is also controlled by a centre
manifold, showing that nonlinear terms dominate the
approach to equilibrium as well as the start of the reaction.
This accounts for the extremely slow and unusual approach to
equilibrium seen in the experimental results of Walde et al.8
Our results show that it is algebraic and not exponential
decay which governs the time evolution in this region :
u(t) B
C
D
D ] D (o [ v )j 1@(j~1)
0
2
0
Kv (j [ 1)(t [ t )
0
2c
D ] D (o [ v )j 1@(j~1)
Lv
0
2
0
0
(8.7)
v(t) B v [
0 (1 ] L v ) Kv (j [ 1)(t [ t )
0
2c
0
Here, v satisÐes
0
Bv
0
Ka (o [ v )K \
,
(8.8)
1
0
D ] D (o [ v)j
0
2
and is the value of v at equilibrium.
Region IV . In the Ðnal approach to equilibrium, a slightly
di†erent set of terms control the kinetics. In this region, more
analytical progress can be made in understanding the
dynamics. The approach to equilibrium occurs on a slow
timescale, with the vesicle concentration (v) slaved to the total
C
(8.1)
B D
KKa vjo2j 1@K
1
t
B
The region has an abrupt end, as t ] t , where
2c
B
1@K
2 o2b (i ] b@) 1@K
2
B
(8.5)
t B
2c
vjo2jKa K
o2 4jia a@vj
1
1
As this time is approached, both u and v appear to blow up.
At this stage the rapid reaction starts, and we need to revert to
the original timescale to examine the kinetics in proper detail.
Region III. This covers the rapid reaction region. Unfortunately the equations are too complex to solve here ; hence
we only have approximations at the start and the end of the
region. At the start, a centre-manifold is identiÐed, showing
the dynamics are essentially nonlinear :
u(t) B o [
u5 \ v(o [ u) ]
B \ b b K(i ] b@)
2 3
K \ 21~jjia@b
3
D \ b (j ] 1)(i ] b@)
0
3
D \ 2~ja@[(2j ] 1)i ] (3j ] 1)b ]
1
3
D \ a (2j ] 1)(i ] b@)
2
2
A
Bvo 1@K
Z~1
K
KKa oj
1
Ka (D ] D oj)uK
1
v(t) B 1 0
B
u(t) B
8 Solution of the contracted kinetic equations for
self-reproducing vesicles
D
D
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
241
concentration u. Physically this corresponds to the two quantities remaining in self-equilibrium with each other throughout
the duration of the region :
C
D
v
1@(j~1)
ev(j~1)(t~C) [ Kv /D
0 =
v
1@(j~1)
L
vBv 1[
0
1 ] L v ev(j~1)(t~C) [ Kv /D
0 =
0
uBo[
G
C
(8.9)
D H
(8.10)
where D \ D ] D (o [ v )j.
=
0
2
0
As a result of this analysis, we Ðnd that our solution has
two regions before the main reactionÈone short region where
one variable reaches a pseudo-steady-state value followed by a
second, long region where it remains slaved to the second
variable. This is the long induction period during which very
little appears to happen. Region III is where the rapid reaction occurs. Even after ignoring terms in the asymptotic
expansion, the equations in this region are too complex to
solve. Thus all we can manage is to write down an approximation for the start of the region and another approximation
for the kinetics at the end of the region. This shows an
unusually slow approach to equilibrium, and undergoes a
slight change as a fourth, Ðnal region is entered, describing the
Ðnal stage of reaching equilibrium which also occurs over a
long timescale.
It must be recalled that all these results are obtained under
the assumption stated in eqn. (7.1), namely that the three types
of vesicles which are deÐned in the contracted BeckerÈDoŽring
scheme (small and pure, large and pure, large and mixed, the
latter implying that anhydride molecules are adsorbed) are
always in thermodynamic equilibrium with one another. The
concentrations of mixed vesicles and large vesicles are then
directly dependent on the concentration of small vesicles, that
is they are slaved by eqn. (7.2). For this to be the case, the
timescale over which they self-equilibrate must be much faster
than any of the other processes concerned.
The behaviour that results from typical choices of parameters is shown in Fig. 6 and can be seen to be in good agreement with the experimental observations of the hydrolysis of
caprylic anhydride by Walde et al. ;8 see also Mavelli and
Luisi.24 With nine parameters in our model and very little
quantitative experimental data, it is impossible to Ðt parameters in a reliable manner. The values chosen are provided
simply to show that our model supports the observed behaviour.
All the analysis presented thus far assumes the initial conditions x (0) \ 0 \ x (0) \ x (0) \ y, s \ o/2. From the deÐni1
2
3
tions of u, v together with eqn. (8.4) and (A2) below, the
monomer concentration is given by
x (t) B vot
(8.11)
1
once a short self-equilibration region is passed (region I). Thus
the time taken for the monomer concentration to reach a
small quantity d is d/vo.
If the system is now initiated with x(0) \ d and s(0) \ o/2,
the total mass is o ] d and the induction time will be shortened by d/v(o ] d). The formula for the induction time is then
modiÐed from (8.5) to
C
D
B
1@K
d
[
(8.12)
KvjKa (o ] d)2j
v(o ] d)
1
This quantity drops to zero if d is sufficiently large, indicating
the existence of a critical aggregation concentration discussed
earlier. Noting that v @ 1, the c.a.c. is
t
ind
B
x
242
1, c.a.c.
\d
c.a.c.
B
A
B
Bv
1@K
Ka Koj~1
1
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
(8.13)
Fig. 6 Graph of total surfactant concentration vs. time, showing the
induction time and slow approach to equilibrium. Parameters :
v \ 0.012, o \ 1.0, j \ 20, D \ 0.01, D \ 1.99, D \ 0.1, B \ 11.976,
0
2 (8.8). The circles
K \ 2.395 and a Ðxed by taking
v \1 0.68 in eqn.
1
represent experimental
results for the0 hydrolysis of caprylic anhydride
as shown in Fig. 6(A) of Walde et al.8
8.2 Centre-manifold perturbation approach
A search for critical points of the unperturbed system [eqn.
(8.1) with v \ 0] leads to two such points : u \ v \ 0, which is
the initial condition, and u \ o, v \ v , which is the equi0
librium conÐguration of the system. In Appendix C we
analyse each of these in turn using the perturbations of the
centre-manifold. This is the same technique we used to solve
the kinetic equations governing the formation of selfreproducing micelles in a previous paper.15 Here we conÐne
ourselves to making a few remarks about the physicochemical
interpretation of this mathematical solution, which provides
additional insights to those obtained from the foregoing
asymptotic analysis.
The fact that there is a centre-manifold emanating from the
equilibrium conÐguration shows that the vesicle distribution is
only just stable, and that a minor structural perturbation to
the system could cause the equilibrium solution to become
unstable or change form dramatically. This behaviour is
gratifying, in view of the known relative instability of many
vesicles. Moreover, the presence of a centre-manifold results in
a very slow approach to equilibrium. States on the centremanifold evolve extremely slowly and so, over short time
intervals, they appear to be at equilibrium. In mathematical
parlance, such states are often called metastable since, if perturbed, they quickly return to a quasi-static state, and then
continue to evolve towards equilibrium only very slowly. This
kind of behaviour is noted in experimental systems as well as
in our model.
9 Discussion
We have proposed a detailed mechanism for vesicle formation
and self-reproduction based on a substantially generalised
version of the BeckerÈDoŽring kinetic scheme. The full scheme
is summarised diagrammatically in Fig. 2 and 3. To our knowl-
edge, this is the Ðrst model for the mechanism of formation of
vesicles which takes into account the stepwise processes by
means of which these structures self-assemble and then reproduce. Previous highly simpliÐed models have assumed the
spontaneous emergence of fully-formed vesicles at the critical
aggregation concentration, and the omnipresence of thermodynamic equilibria between monomers, micelles and
vesicles.24,30
In order to make our general model analytically tractable,
as well as to eliminate the need to determine or Ðt a very large
number of generally unknown rate coefficients for the individual molecular processes comprising the full reaction scheme,
we have developed a systematic contraction or coarsegraining approximation procedure which reduces the dimensionality of the kinetic equations while making them much
more suitable for direct comparison with macroscopic data.
This leads to a relatively simple macroscopic description of
the system.
The speciÐc experiments and the model we have proposed
here have possible relevance to the chemical origins of life,31
since the chemical processes involved provide a direct route to
the formation of bounded cell-like structures under prebiotic
conditions. The BeckerÈDoŽring model gives results that agree
well with experimental data, at least in part because it
describes the existence of a critical aggregation concentration
in a natural way. One manifestation of this is shown by the
dependence of the induction time on the initial surfactant concentration : when this reaches the c.a.c., the induction time falls
to zero.
It is possible to include more general aggregation and fragmentation processes than the one-step monomer attachment
and detachment kinetics implied by the BeckerÈDoŽring
scheme. Here, this would considerably complicate an already
difficult theoretical problem. However, we hope to return in
the future with an analysis of some related coagulationÈ
fragmentation problems involving surfactants in which such a
generalisation can be fruitfully studied.25
We are grateful to Pier Luigi Luisi, Peter Walde, Kenichi
Morigaki, Neville Boden, Richard Harding, and John Billingham for several helpful discussions, and to Marco Maestro
for making available to us a preprint of ref. 24. P.V.C. is grateful to Luigi Luisi for an invitation to visit the Institut fuŽr
Polymere at E.T.H., ZuŽrich, in June 1996 and for his kind
hospitality ; also to Wolfson College and the subdepartment of
Theoretical Physics at the University of Oxford for a Visiting
Fellowship. J.A.D.W. is grateful to the Nuffield Foundation
for the provision of computing equipment.
Appendix A : the function Z (s)
n
In ref. 15 we introduced the function Z (s) deÐned by
n
s dx
(A1)
Z (s) \
n
1 ] xn
0
For convenience, we summarise its properties again here. For
small s, the integrand can be expanded around x \ 0 to Ðnd
P
A
B
1 n]1
, [sn
Z (s) \ s F 1, ,
n
2 1
n
n
Bs[
sn ] 1
s2n`1
]
]ÉÉÉ
n ] 1 2n ] 1
(A2)
where F (a, b, c, x) is a hypergeometric function. Since the
2 1
value of n that concerns us is large, this can be approximated
by F (1, 0, 1, x) \ 1.
2 1
For large s we note that the quantity Z (O) can be calcun
lated by contour integration in the complex plane [Z (O) \
n
n/n cosec(n/n)]. Further terms in the series for large s can be
obtained by subtracting the contribution from (s, O) (the sum-
mation only converges for s [ 1)
AB
A
B
n
n
s1~n
n [ 1 2n [ 1
Z (s) \ cosec
[
F 1,
,
, [s~n
n
n
n
n[1 2 1
n
n
(A3)
Again, since our value of n is large, the hypergeometric function can be approximated by
F (1, 1, 2, x) \
2 1
[x~1 log(1 [ x) [Abramowitz and Stegun,32 eqn. (15.1.3)].
Appendix B : solution of eqn. (8.1) by matched
asymptotic expansions
We use subscripts on the u, v variables to denote the scaled
solution in each region.
B1 : region I
The initial conditions are u(0) \ 0 \ v(0). Thus we seek a
scaling for u and v which make both small, this is consistent
when u B vu , v B vKv , t B v0q , so that our leading order
1
1
1
equations are
Bv
1
u5 \ o, v5 \ Ka uK [
(B1)
1
1 1 D ] D oj
1
0
1
where D , D are as deÐned in eqn. (8.2). The solution is
0 1
t
(B2)
u \ ot, v \ Ka oK e~ht ehssK ds
1
1
1
0
where h \ B/(D ] D oj). This solution will break down when
0
1
u B 1, or when u B v, or when v B v. The Ðrst two occur at
t B v~1, the latter at t B v~j@K @ v~1. When t reaches this
second value, u B v1@K, and we need to start a new region.
P
B2 : region II
The correct scaling in this region is as described above ; we
put u B v1@Ku , v B vv and t B v~j@Kq . The leading order
2
2
2
equations are then
Bv
2
(B3)
0 \ Ka uK [
1 2 D ] D oj
0
1
From the latter, we see that v is slaved to u :
2
2
a (D ] D oj)uK
1
2
v \ 1 0
(B4)
2
b b (i ] b@)
2 3
The former eqn. (B3) can now be solved to provide an implicit
solution
u@ \ o ]
2
q \
2
P
KoKKa v
1 2,
B
u2
AB A CD B
du
1 o 1@K
g 1@K
\
Z u
K 2 o
o ] guK o g
(B5)
0
where g \ KKa oj/B. See Appendix A for details of the func1
tion Z (s). Since Z (s) tends to the constant Z \ n cosec(n/n)/n
n
n
as s ] O, our solution blows up (u ] O) in Ðnite time, (as
2
q ] q ), where
2
2c
2 o2b (i ] b@) 1@K
2
(B6)
q B
2c o2
4a@a ji
1
since in our applications K is large, so (n/K)cosec(n/K) B 1.
Owing to the long timescale used here, region II is the long
induction period and has an abrupt ending. Formally the
solution is
C
A BA
D
B
A CD B
G A C D BH
b o2(i ] b@) 1@K
g 1@K
2
Z~1 oq
K
2 o
4a a@ij
1
g 1@K K
2j~1D o
0
v (q ) \
Z~1 oq
2 2
K
2
o
a@b ojji
3
2
u (q ) \
2 2
o
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
(B7)
243
At the end of region II, both u , v ] O, as q ] q , with
2 2
2
2c
v B uK , from which, we can Ðnd the scaling for the next
2
2
region : u B 1 B v with the initial condition that u and v are
both small and satisfy v P uj .
2
2
The induction time is similar to which occurred in our modelling of generalised nucleation theory ;21 the underlying equations have the same mathematical structure, a long induction
region caused by one (fast) variable being slaved to another,
which is varying extremely slowly.
B3 : region III
Following the conditions at the end of region II, we expect
that in region III both u and v are O(1) and vary by O(1)
amounts when t varies by an O(1) amount. A direct asymptotic expansion of eqn. (8.1) then leads to the set of nonlinear
equations
Kv(o [ u)j
D ] D (o [ u)j ] D (u [ v)j
0
1
2
Bv
v5 \ Ka (u [ v)K [
(B8)
1
D ] D (o [ u)j ] D (u [ v)j
0
1
2
which involve too many terms to be solvable. Hence we shall
consider only how the kinetics accelerate and slow down at
the start and end of region III.
u5 \
Start of region III. Firstly, let us consider just the start of
region III. Since u B v1@K and v B v in region II, our initial
conditions in region III are going to be small and hence close
to the critical point at u \ 0 \ v.
Linearising about the origin, we Ðnd the eigenvalues are
zero and [B/(D ] D oj), indicating the existence of a centre0
1
manifold which is itself stable. To Ðnd the centre-manifold, we
need to introduce w \ Bu ] Kojv in place of u to form a
system whose linear part is diagonal. The centre-manifold is
then found in a way similar to that presented in Section C1
(except that here we do not have to worry about v-corrections
to the centre-manifold). Thus the centre-manifold is given by
vB
A
B
D oj
Ka
1 D ] 1
wK
0
B
BK
(B9)
to leading order. On the centre-manifold, the equation of
motion is
AB
w K
w5 \ KKa oj
1
B
which is solved by
A
(B10)
B
BK
1@j
(B11)
BKw~j [ jKa ojKt
0
1
Then v is found from eqn. (B9) and u from u \ (w [ Koj v)/B ;
see eqn. (8.6) for the formulae.
Since we know u \ o, this solution breaks down owing to
the presence of further nonlinear terms, which are neglected in
the above analysis. It is not possible to Ðnd a solution valid
for the entirety of region III, but we can Ðnd the form of the
solution as uˆo, as we shall now show.
w(t) B
End of region III. In this region we introduce u \ o [ u,
3
v \ v [ v and treat both of these as small but larger than v.
3
0
Both are positive but approaching zero. The leading order
terms in the equations for u are
3
Kv uj
0 3
u5 \
3 D ] D (o [ v )j
0
2
0
v
[Bv
3 ] L (v [ u )
0
(B12)
v5 \
3
3
3 D ] D (o [ v )j v
0
0
2
0
C
244
D
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
Once again we see that nonlinear terms dominate the behaviour of the system through the existence of a centre-manifold.
To diagonalise the linear part of the system the variable w \
3
v /v ] L (v [ u ) is used to eliminate v . We then Ðnd
3 0
3
3
3
w5 \ [Pw [ Su2
(B13)
3
3
3
with P and S as in eqn. (C17). The centre-manifold is given by
w \ [Su 2/P. To Ðnd the kinetics we Ðrst Ðnd u from (B12) :
3
3
3
D ] D (o [ v )j 1@(j~1)
0
2
0
(B14)
u (t) B
3
Kv (j [ 1)(t [ C)
0
This is only valid for t [ C ; hence to Ðnd a reasonable
approximation to the solution we put C \ t .
2c
Inverting the above transformations we Ðnd
C
D
(w ] L u )v
Lv u
3
3 0B
0 3 ] O(u2)
(B15)
3
1]L v
1]L v
0
0
hence u and v both tend to zero at the same rate (with their
3
3
ratio being constant).
v \
3
B4 : region IV
When uj B vu , a new balance in eqn. (8.1) is obtained, and a
3
3
new region entered. In this region, we again have a long timescale q \ vt, with u \ o [ v1@(j~1)u and v \ v [ v1@(j~1)v .
4
4
0
4
The leading order equations in this region are
Kv uj
v
0 4
u@ \ [u [
, 0 \ 4 ] L (v [ u ) (B16)
4
4 D ] D (o [ v )j
4
4
v
0
2
0
0
Once again we have a slaved equation since the system is
evolving on a slow timescale the vesicles are always in equilibrium with the monomer. The Ðrst equation above is a Bernoulli equation, which can be integrated by separating
variables to Ðnd
1
(B17)
u \
4 [e(j~1)(q4~C) [ Kv /D ]1@(j~1)
0 =
where D \ D ] D (o [ v )j. The second equation then
=
0
2
0
implies v \ L v u /(1 ] L v ) ; as noted in the previous section,
4
0 4
0
the quantities (o [ u) and (v [ v) both tend to zero with a
0
constant ratio.
Appendix C : centre-manifold perturbation solution
of eqn. (8.1)
As stated in the main text, a search for critical points of the
unperturbed system [eqn. (8.1) with v \ 0] leads to two
points : u \ v \ 0, which is the initial condition, and u \ o,
v \ v , which is the equilibrium conÐguration of the system.
0
C1 : near the initial condition (u = ¿ = 0)
Ignoring the O(v) terms, the linear part of the system at the
origin can be written as
AB
A
BA B
u5
0 Koj u
1
\
(C1)
D ] D oj 0 [B v
v5
0
1
which clearly has a zero eigenvalue (the other is negative, indicating that the centre-manifold is stable and hence worth
studying). We transform the variables to (v, w) from (u, v) via
w \ Kojv ] Bu 7 u \
w [ Kojv
B
(C2)
so as to end up with a system whose linear part is diagonal.
Then the centre-manifold can be written as
v \ h(w) B h wP ] h wQ ] vh wR
0
1
v
(C3)
and the exponents P, Q, R and coefficients h , h , h are found
0 1 v
from substituting the asymptotic forms for v5 , w5 into v5 \
5
h@(w)w :
AB
w K jKoj~1wv
[
w5 \ Bvo ] KojKa
1 B
D ] D oj
0
1
w K
Bv
jD oj~1wv
1
[
[
(C4)
v5 \ Ka
1 B
D ] D oj (D ] D oj)2
0
1
0
1
Balancing terms gives P \ K, Q \ K ] 1 and R \ K [ 1, so
that the centre-manifold is given by
AB
v \ h(w) B
C
A
B
Ka D ] D oj
0
1
1
wK
B
BK
D
P
A
B CA
B D
C
D
Bvo(vot)K
(C7)
KojKa (K ] 1)
1
The form of this function is a slow linear growth from
w(0) \ 0 followed by a rapid take-o† at larger times caused by
the second term in the square brackets.
Using eqn. (C5) we can now Ðnd v(t) ; to leading order this is
w B Bvot 1 ]
v(t) B
A
B
Ka D ] D oj
0
1 (Bvot)K
1
B
B
(C8)
Although introduced to simplify the mathematics, this quantity represents the mass of vesicular material in the system,
and is hence an important quantity. It remains small for a
long period of time before increasing dramatically, as was
observed with the amount of micellar material in the ethyl
caprylate system studied previously.15
Having found w and v, we are in a position to calculate u.
As with v this was introduced to simplify the calculations, but
also has a direct relationship to the original system. It represents the total surfactant present in the system : u \ x
1
] Kx ] (K ] j)x ] (K ] 2j)y \ o [ 2s. It is this quantity
2
3
which is plotted in Fig. 4 of Mavelli and Luisi.24 From eqn.
(C2), we Ðnd
C
u(t) B vot 1 [ KojKa (Bvot)j
1
A
BD
D ] D oj
0
1
B
(C9)
This has the correct form expected : it displays a long period
of linear growth in time, followed by a Ñattening out. A simple
calculation shows that it reaches a zero gradient at
C
D
1
~1@j
1
KojK2a (D ] D oj)
t \
1 0
1
c Bvo B
Kuj(v ] v )
1 0
1
u5 \ [vu [
1
1 D ] D uj ] D (o [ v [ u [ v )j
0
1 1
2
0
1
1
[Bv
u ]v K
0
1
v5 \
1[ 1[ 1
1 D ] D (o [ v )j
o[v
0
2
0
0
B[D v uj[D v ]D v (o[v [u [v )j[D v (o[v )j]
0 0 1
0 1
2 0
0
1
1
2 0
0
]
[D ] D (o [ v )j][D ] D uj ] D (o [ v [ u [ v )j]
0
2
0
0
1 1
2
0
1
1
(C11)
C A
BD
The only linear part of this system is in the v5 equations,
1
where an expansion around the equilibrium solution yields
vKo(D ] D oj)
jD oj~1w
0
1 [
1
(C5)
w
(D ] D oj)
0
1
The equation of motion on the centre-manifold near to the
critical point is w5 \ Bvo ] KojKa (w/B)K. This can be inte1
grated using the function Z (s) detailed in Appendix A to give
n
w@B
B ds
t\
Bvo ] KojKa sK
1
0
1
Bvo 1@K
w KojKa 1@K
1
\
Z
(C6)
K B
vo KojKa
Bvo
1
Inverting this expression for small w leads to
] 1[
v \ v [ v , in which the system of equations can be written
1
0
as
(C10)
where u B vojt /K.
c
C2 : near equilibrium (u = q, ¿ = ¿ )
0
We translate the system so that the equilibrium solution is at
the origin, and we work with the new variables u \ o [ u,
1
C A
A
C
B
v
u ]v
1
v5 B [Ka (o [ v )K 1 ] 1
1
1
0 v
o[v
0
0
jD (o [ v )j
2
0
] K]
D ] D (o [ v )j
0
2
0
v
\ [Ka (o [ v )K 1 ] L (u ] v )
1
1
1
0 v
0
where we have deÐned the new constant
A
BC
BD
D
(C12)
D
1
jD (o [ v )j
2
0
K]
(C13)
o[v
D ] D (o [ v )j
0
0
2
0
As in the previous calculation, we use this to transform coordinates so that the linear part of the system is diagonal. To do
this we introduce
L\
v
(w [ L u )v
1 0
w \ 1 ] L (u ] v ) 7 v \ 1
1 v
1
1
1
1]L v
0
0
u ]v w
0 1
u ]v \ 1
1
1
1]L v
0
Then our equations can be written as
(C14)
u5 \ vu [ K1 uj
(C15)
1
1
1
w5 \ [L vu [ Pw [ Su2 [ Qw2 [ Ru w [ É É É
(C16)
1
1
1
1
1
1 1
for constants P, Q, R, S, K1 , of which the most important are
Kv
0
D ] D (o [ v )j
0
2
0
B(1 ] L v )
0
P\
D ] D (o [ v )j
0
2
0
1
S\
2(o [ v )
0
[ jB[KD ] 2D (K ] j)(o [ v )j]
0
2
0
]
[D ] D (o [ v )j][(o ] jv )D ] (o ] 2jv )D (o [ v )j]
0
2
0
0 0
0 2
0
(C17)
K1 \
Using the perturbed centre-manifold approach, we seek a
function
w \ h(u ) B h un ] É É É ] vh um ] É É É
(C18)
1
1
0 1
v 1
with exponents n, m and constants h , h such that the leading
0 v
order terms in the system (C15)È(C16) balance. Firstly, ignoring the O(v) terms, we Ðnd n \ 2, and h \ [S/P ; then from
0
the O(v) terms we Ðnd m \ 1 and h \ [L /P. Thus the perv
turbed centre-manifold is
[1
w B
(Su2 ] L vu )
1
1
1
P
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
(C19)
245
Now that we know where the centre-manifold is, we are in a
position to Ðnd the form of the kinetics which occur on it.
These are found by solving eqn. (C15), an equation which only
involves u . We know that u ] 0 as t ] ]O, but the solu1
1
tion has a divergence at Ðnite time. We impose the boundary
condition that this divergence occurs at t \ t
, giving
rapid
v
1@(j~1)
(C20)
u \
1
K1 Mexp[v(j [ 1)(t [ t
)] [ 1N
rapid
From eqn. (C17), S \ 0, we see that w [ 0, but more impor1
tantly eqn. (C19) implies w @ u . Thus when we calculate v
1
1
1
from eqn. (C14) we Ðnd
A
B
[L v u
0 1
v B
1 1]L v
0
v (1 ] j)D ] v (1 ] 2j)D (o [ v )j
0
0
2
0
B[ 0
(o ] jv )D ] (o ] 2jv )D (o [ v )j
0 0
0 2
0
v
1@(j~1)
(C21)
]
K1 Mexp[v(j [ 1)(t [ t
)] [ 1N
rapid
Thus both the total surfactant (u) and total vesicular mass (v)
approach equilibrium in the same manner.
C
A
D
B
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Paper 7/0348K ; Received 20th May, 1997