J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1953-1959 Kinetics of Self-replicating Micelles J. Billingham and P. V. Coveney Schlumberger Cambridge Research, High Cross, Madingfey Road, Cambridge, UK CB3 OHG Recent experiments which achieved the autopoiesis of caprylate micelles by aqueous alkaline hydrolysis of ethyl caprylate have shown that the kinetics of this process are highly non-linear. There is an extended induction period during which the concentration of micelles remains small; at the end of this period, ethyl caprylate is consumed and micelles form rapidly via an autocatalytic reaction. In this paper, we investigate two macroscopic kinetic models for this process. In the first, we specifically include the equilibrium critical micellar concentration for caprylate anions and require that self-replication of micelles only occurs beyond this point.In the second, we study the properties of a non-linear model which is capable of accounting for the very sharp growth in total caprylate concentration as a function of time, without making any assumptions concerning equilibrium. Although both of these models have limitations, they provide insight into what features a more realistic model must have. We conclude by formulating a mesoscopic model that provides a much more detailed description of the micro- scopic processes of micelle formation, and which observed react ion kinetics. 1. Introduction The synthesis of autopoietic systems in the laboratory is cur- rently under investigation.? The motivation for this work originates in a definition of the minimal conditions necessary for a system to be called living, proposed by Maturana and Varela.' In brief, they define autopoiesis as a process whereby a system which possesses well defined boundaries is capable of self-replication.It can be argued that the micellar system that we shall describe here is indeed such an autopoietic system and therefore can be regarded as fulfilling this cri- terion for minimal life.2,3 In the present paper, we shall study the kinetics of this system in some detail. The most obvious feature of the reaction is that there is a well defined induction time prior to very rapid growth in the production of micelles. We begin by briefly summarising the experimental techniques employed, and then describe our theoretical analysis of the non-linear kinetics of self-replication.We conclude by outlin- ing a mesoscopic approach which may provide a more realis- tic model of the kinetics of micelle formation than the macroscopic models that we describe here. 2. Experimental Methods and Observations The basic experimental set-up is very simple. Immiscible ethyl caprylate and a concentrated (3 mol dm-3) aqueous solution of sodium hydroxide are stirred at 150 rpm and heated under reflux, typically at temperatures close to 100°C. Any alcohol produced in the reaction boils off under reflux. The concen- tration of ethyl caprylate and the total caprylate anion concentration are conveniently determined by FTIR spec-troscopy. The concentration of the developing micellar struc- tures is determined from time-resolved fluorescence q~enching.~ It is found that for a well defined period after the start of reaction very little activity takes place, as measured by changes in the concentrations of the aforementioned species.However, at the end of this quiescent period the reaction sud- denly takes off at a dramatic rate: all of the ester is consumed and the production of micelles proceeds apace, as illustrated in Fig. 1. The final state of the system, after about 34 h, is a homogeneous clear liquid. Experimentally, the induction A group at the Institut fur Polymere at the E.T.H. in Zurich is actively researching this area.' we believe will provide a quantitative description of the period is found to depend strongly on both the temperature and the initial concentration of caprylate ions.An important quantity is the critical micellar concentration (c.m.c.) of the caprylate surfactant system. The c.m.c. is defined as the concentration of monomers above which micelles are formed. Note that this is an equilibrium property of a micellar system, and therefore not necessarily relevant to the experiments currently under discussion. In addition, it is notoriously difficult to measure c.m.c.s unambiguously, not only because the results are strongly dependent on which technique is used for making the measurements, but also because micelle clusters are actually present at low concentra- tions below the notional c.m.c.One simple way of measuring c.m.c.s makes use of the cloud point in a surfactant system as the concentration of monomers is steadily increased. For the surfactant described above, the c.m.c. was found by this method to be 0.1 mol dm-3, under the conditions pertaining in this experiment. The number of caprylate monomers per micelle determined experimentally is roughly n = 63. The ester dissolves in the micelles, where it is rapidly hydrolysed to form more sur- factant and hence micelles. It is apparent that the micelles are self-replicating insofar as they act as a cross-catalyst for the hydrolysis of ester in the bulk of the aqueous phase, rather than being restricted to the macroscopic ester/water interface as would be the case in their absence. The aqueous caprylate micelles formed can be reversibly transformed into vesicles by decreasing the pH to 6.5.These have been observed using freeze-fracture electron microscopy, and have a radius of around 150 nm. The same effect can be achieved by bubbling CO, through the solution. Note also that use of caprylic anhydride in place of the ethyl caprylate ester leads to hydrolysis at lower pH, whereby vesicles are formed directly. 3. Theoretical Analysis In purely qualitative terms, it is simple to account for the behaviour of this system, as described in the previous section. However, it is of considerable theoretical importance to be able to furnish a more detailed account of the processes involved in this example of autopoiesis.For example, from the point of view of prebiotic chemistry one would like to be able to furnish estimates of induction times and timescales for the period of rapid reaction. The modelling approach described in this paper also serves to highlight the basis for the unusual time-dependent properties of self-assembling supramolcular micellar systems in general. As such, it should be of general interest inter alia to those working on the kinetics of self-replicating systems.' Let EC represent ethyl caprylate, C, the caprylate anion monomer, and C,, the micelle formed from caprylate anions, assumed as in ref. 2 to be monodisperse with n = 63. From a theoretical standpoint, the assumption of monodispersity is, as will become clear in this work, a rather strong one.In broad terms, the following processes occur, EC is con- verted to C via two possible routes. In the first, the step EC -P C occurs directly through alkaline hydrolysis in the presence of excess hydroxide. The C produced in this manner can then form C, in a process which, if it were at equilibrium, would only proceed if the concentration of C exceeds the c.m.c. Once Cnstarts to appear in the system, unreacted EC is solubilised within the micelles, and is then hydrolysed. If this last process is fast, its effect is to cross-catalyse the pro- duction of C (and, of course, indirectly C, itself) thereby catalysing the production of micelles. Overall, therefore, the production of micelles is autocatalytic in that, once some are formed, they accelerate their own rate of formation.In the following subsections we analyse two models for the kinetics of micelle formation. Although we shall see that neither of these models is entirely satisfactory, an understand- ing of the difficulties that arise in connection with each scheme tells us a lot about the features which a more com- plete model should have. 3.1. Model with known C.M.C. The reaction scheme for this model is rate = 0; for CCl [Clc.rn.c. C, + EC C, + C; rate = k2[EC][Cn] (14 Here, square brackets denote the concentration of the given chemical species. The c.m.c. is written as [C]c.m.c,.The three steps in reaction (I) represent : uncatalysed hydrolysis of ethyl caprylate, EC, to produce the caprylate monomer, C, by pseudo-first-order kinetics, assuming that this takes place in the aqueous phase where hydroxide ion is present in large excess compared to the ester [reaction (Ia)]; formation of a micelle, C,, from n caprylate monomers, C [reaction (Ib)]; and hydrolysis of ethyl caprylate, EC, to produce the capry- late monomer, C, catalysed by the presence of micelles, C, [reaction (Ic)]. The uncatalysed formation of micelles directly from capry- late monomers is assumed to be possible only when the caprylate monomer concentration exceeds the c.m.c.([C] > [C]c.m.c.).Until this occurs, the only reaction which proceeds is the simple pseudo-first-order decomposition of the ethyl caprylate. Note that these processes are not intended to rep- resent the stoichiometric reactions themselves.For example, we are not suggesting that n caprylate monomers combine simultaneously to produce a micelle. The second reaction step is a simple representation of the overall uncatalysed process of micelle formation. Note that our assumption of pseudo-first-order kinetics is also justified by the close agree- ment that we obtain with the experimental results. It is clear J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 that for 0 < t 6 t,,,.,., where Eqn. (1) and (2)tell us that the caprylate monomer concentra- tion, [C], grows until t = tc.m.c.. When t = t,,,.,., the capry- late monomer concentration reaches the c.m.c. ([C] = [C],.,.,), and the remaining reaction steps start to proceed.The results of the experiment, where tc.m.c,x 33 h, [C]c.m.c.= 0.1 mol dm-3, n z63 and [EClo z 1.425 mol dm-3, suggest that k, 6.1 x s-'. These values are all deduced from Fig. 1. Note that, because the ester and aqueous alkaline solution together comprise a biphasic system, the interfacial area between the immiscible fluids is an important parameter influencing the numerical value of the rate constant k,, which can thus be expected to vary from one experimental set-up to another if different volumes of the phases or rates of stirring are used.6 For t > tc.m,c,,the reaction rate equations are dCEC1--k,[EC] -k,[C,][EC]dt These equations are non-linear, and do not have a closed- form solution. It is, however, straightforward to obtain a numerical solution using the fourth-order Runge-Kutta method (see, for example, ref.7). We can simulate the experi- mental results shown in Fig. 1 by using the parameter values: k, = 6.1 x s-l, k, = s-l, k, = 0.19 dm3 mol-' s-l, [C]c.rn.c. = 0.1 mol dm-3, [EC], = 1.425 mol dm-3, n = 63. This value of k, has been determined experimentally, and is given in ref. 2. The size of k, determines the rate at which micelles are produced once the caprylate concentration exceeds the c.m.c. The value given above leads to good agree- ment of the model with the experimental results. The exact 1.5-* 25 1U-EUE" 1.2 ----20 5 E0 E 4---.2 0.9--15 s cC .-c 0 c .-.-. I.-6 0.6-'0 2 4-s0.3---5 -w-is =:JP -w 0zo.0-: : : -: .: . : ' i-0 'E J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 value of k, is not critically important, since any value of this order of magnitude gives similar results. The predicted micelle and total caprylate concentrations are illustrated in Fig. 2 and 3 for [C], = 0 and [C], = 0.02 mol dm-3, respectively. These should be compared with Fig. 1, which shows the experimental results. The observed and theoretical results are in good agreement. At first sight, we appear to have a very successful mathe-matical model for the kinetics of micelle formation. The problem is that, because the model is by nature very largely empirical, we have actually learnt very little. We have con-firmed that it is possible to model this autopoietic system in a piecewise fashion by assuming that ethyl caprylate is hydro-lysed by hydroxide ion to produce caprylate monomers until the c.m.c.is reached, when micelles start to form. These then rapidly catalyse their own production, thereby exhausting the supply of the ester. However, by adopting this piecewise approach, our understanding of the coupled nature of the overall process has not been addressed. At the heart of the problem is the phenomenon of a well defined c.m.c., which this model makes no attempt to explain, but simply takes for granted. The requirement that the pro-duction of further micelles, and the hydrolysis of ester within such micelles, may occur only when [C] > [C]c,m.c, is undoubtedly only a first approximation.It is not clear a priori whether the rate of micellar-catalysed hydrolysis is slow compared to the rate at which micelles are formed, itself a non-equilibrium process for which the assumption of equi-librium is merely an approximation. The assumption of a thermodynamic equilibrium for one or more steps within a given kinetic scheme is a familiar one s Oa6 t V -time/h Fig. 2 Total caprylate (---) and micelle (-) concentrations when no caprylate is present initially, predicted from the model of reaction (I) with k, = 6.1 x lo-' s-', k, = lop4 s-', k, = 0.19 dm3 mol-' s-l, [C]c,m,c,= 0.1 mol drnp3, [EC), = 1.425 mol dmF3, n = 63 1 E 1.5 i. I ' ,--L-__--___--'L_-_-_---------------_25 7 ,U E-;1.2 ---20 5 u .. E 0 --.E'= 0.9 --152 .-c -C 2 c c2 0.6 --10 6 0 0 Q, --Z 0.3 -5 - :0 2-P 9 o.o ,--_--------7 __________---------Q1 I ,,I, 1.; 0 which is used to simplify the mathematical analysis of the resulting reaction rate equations.It is by no means obvious, however, that such an assumption is generally justified. An alternative approach is to try to formulate a global non-equilibrium model wherein all the steps are coupled and which does not explicitly include a c.m.c., but where the con-centration of micelles remains small for a well defined induc-tion period, after which the concentration grows rapidly. This type of kinetic scheme describes what is known as a clock reaction.Examples of simple model clock reaction schemes, where the induction period arises from the interaction of the various reaction steps rather than by its explicit inclusion in the model, have recently been studied in a series of papers.'.' In the next section, we propose a clock reaction model for autopoietic micelle formation. 3.2. Clock Reaction Model The model that we propose is very similar to the model of reaction scheme (I). It is given by EC --* C; rate = k,[EC] (114 nC +C,; rate = k,[C] (IIb) 2C, + nEC +3C,; rate = k2[EC][CJ2 (IIc) These three simple reaction steps represent : uncatalysed hydrolysis of ethyl caprylate, EC, to produce the caprylate monomer, C [reaction (IIa)] ;formation of a micelle, C, ,from n caprylate monomers, C [reaction (IIb)] ; and autocatalytic production of a new micelle, C,, from n ethyl caprylate mol-ecules, EC [reaction (IIc)].The first step is the same as that in reaction (I), whilst the rate expression for the second is slightly different since we do not explicitly include a c.m.c. The third step requires some explanation. Since in reaction scheme (11) we have not included a c.m.c. explicitly, the uncatalysed production of micelles from monomers must take place very slowly if we are to have a long induction period before a significant concentration of micelles is formed. If we were simply to modify reaction scheme (I) by setting [C]c.m.c, to zero, and making k, much smaller, there would be no mechanism for the rapid production of micelles.This leads us to postulate an autocatalytic step, whereby micelles catalyse their own production, rather than cross-catalysing the pro-duction of monomers, as in reaction (I). Finally, from our previous analysis of model clock reaction schemes, we know that a cubic autocatalytic step is both realistic and leads to a more clearly defined induction period than a quadratic auto-catalytic step (rate = k,[EC][C,]). Note that the autocatalytic step should not be treated as corresponding to an elementary process. We are not suggest-ing that tens of ethyl caprylate molecules interact with two micelles to produce three micelles in a single collisional inter-action. As in many other kinetic schemes, this step should be interpreted as representing the overall effect of a complex sequence of more elementary processes.These more elemen-tary processes are described in more detail in the model we discuss in Section 4. The reaction rate equations for the kinetic scheme (11)are -=dCEC' -k,[EC] -nk,[EC][C,12dt dCcl -k,[EC] -nk, [C] (4)dt dtdccnl -k,[C] + k2[EC][C,]2 1956 As before, although these equations have no closed-form solution, we can obtain solutions numerically using the fourth-order Runge-Kutta method. However, for this set of equations, we can also obtain a useful asymptotic solution for the case where the catalysed hydrolysis of ethyl caprylate proceeds much faster than the uncatalysed hydrolysis (k,[EC]g + k,). We summarise the results here and give more details in the Appendix.We can show that there is then a well defined induction period where the micelle concentration is small and the domi- nant reaction step is the slow, uncatalysed hydrolysis of ethyl caprylate to produce caprylate monomers. Meanwhile the uncatalysed production of micelles from caprylate monomers proceeds very slowly, and the autocatalytic step more slowly still. At the end of the induction period, the autocatalytic step rapidly becomes dominant, and the ethyl caprylate is con- sumed to produce micelles. Once the ethyl caprylate has been consumed, only the very slow, uncatalysed combination of monomers into micelles can proceed. The asymptotic analysis described in the Appendix allows us to determine the duration of the induction period for given values of the parameters. We have chosen the values of k,, k,, [EC], and n to be as in Section 3.1. However, as we explained above, the value of k, needs to be much smaller than before in order to have a long induction period.The asymptotic analysis shows that we must take k, = 2.0 x s-l in order to achieve agreement with the observed 33 h long induction period. Fig. 4 shows the numerical solution of eqn. (4) for [C], = 0, and the other parameter values taken as follows: k, = 6.1 x s-l, k, = 2.0 x s-l, k, = 0.19 dm6 mol-2 s-l, [EC], = 1.425 mol dm-3, n = 63, This numerical solution is consistent with the asymptotic solution, and shows that the model can reproduce the behaviour seen in the experiment when no caprylate is present initially.Note that, although the concentration of micelles appears to be constant in Fig. 4 and 5 once the induction period is over, there is actually an extremely slow uncatalysed conversion of caprylate monomer to micelles. Fig. 5 shows the numerical solution of eqn. (4) for [C], = 0.02 mol dm-3. This shows that the duration of the induction period is reduced to around 28 h, consistent with the asymp- totic solution, as we would expect, but not in quantitative agreement with the experimentally measured induction period of around 25 h. The duration of the induction period, to, predicted by the model as a function of [C],, is shown in Fig. 6, along with the experimental values. We can understand the discrepancy between the model and the experimental observations by investigating whether the -.rL.E" 0.9 t-2 Y 0.6 0.31-Q) I I 10 .E 40 50 60 time/hc, Fig. 4 Total caprylate (---) and micelle (-) concentrations when no caprylate is present initially, predicted from the model of reaction (11) with k, = 6.1 x lo-' s-', k, = 2.0 x lo-* s-l, k, = 0.19 dm6 molP2 s-l, [EC], = 1.475 mol dmP3, n = 63 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 c0 tirne/hFig. 5 Total caprylate (---) and micelle (-) concentrations when [C] = 0.02 mol dm-3 initially, predicted from the model of reaction (11) with the parameters given for Fig. 4 model effectively has a c.m.c. associated with it. Fig. 7 shows the behaviour of the total caprylate concentration, [C],,, = [C] + n[C,], predicted by the model for various initial capry- late concentrations, [C],.Clearly, the value of [C],,, at the end of the induction period varies as [C], varies. We need [C],,, = [C]c.m.c.at the end of the induction period, irrespec- tive of the value of [C],, if the model is to reproduce the experimental observations. It is clear that, in the present model, regardless of how much caprylate monomer is present initially, there is always a finite induction time needed for the uncatalysed step to produce enough micelles for the autocat- alytic step to become significant. 3025 L 4 i 5 1~I I ,,I,, 0 0 25 50 75 100 125 150 175 200 [C],/rnmol dm-3 Fig. 6 Induction time, to, predicted by the model of reaction (11) with the parameters given for Fig.4, as a function of [C],. (A) Experimental values; (M)c.m.c. 1.80 1.60 1.40 m E 1.20 1.00 0.80 21 0.60 o, 0.40 0 10 20 30 40 50 60 time/h Fig. 7 Total caprylate concentration, [C],,,, for various values of the initial caprylate monomer concentration, [C], , predicted from the model of reaction (11) with the parameters given for Fig. 4. [Cl0/mol dm-3: (a) 0, (b) 0.02, (c) 0.06, (d) 0.1, (e) 0.2. The horizontal dotted line represents the expected value of the c.m.c. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 We conclude that the model of reaction (11) is able to reproduce the long induction period observed experimentally, but that it does not agree with the experimentally observed c.m.c., incorrectly predicting that a finite induction period persists for any initial caprylate monomer concentration, [C],.1° One reason for this may be that the model does not include the possibility of micelles being converted back to monomers.4. Mesoscopic Model It should be evident that the models described above are very much macroscopic and semi-empirical in nature. A more detailed, microscopic treatment would be preferable, in which we consider the kinetics of each 'elementary step' in the process of micelle formation. This would enable us to relax many of the assumptions invoked in Section 3, including that of the monodispersity of micelles. Such an approach has recently been proposed by Carr et al.for the study of multi-component micelle kinetics in the absence of chemical reac- tions." These authors argue that the well known Becker-Doring equations (which provide the theoretical basis underpinning the generally very successful classical nucle- ation theory" and the theoretical description of the kinetics of a variety of thermodynamic phase transitions) may be used to study the process of micellization under non-equilibrium conditions.' Moreover, the long-time properties of this system provide a description familiar from equilibrium ther- modynamics." We are not aware of any other attempt to model micellar kinetics in this fashion, although some earlier and less general work was based on the same fundamental approximations.l4 Strictly speaking, the Becker-Doring theory is mesoscopic rather than microscopic, for reasons we shall explain shortly. The basic assumption of the Becker-Doring theory is that clusters of particles (in this case micelles) change their sizes through a series of one-step processes in each of which a given cluster grows or shrinks by one monomer molecule at a time. The general kinetic process in Becker-Doring theory can be written as with r ranging from one to infinity. Here a, and b,+l are the rate coefficients for the forward and reverse steps, respec- tively. The reaction rates are given by the law of mass action, and lead to an infinite system of coupled non-linear differen- tial equations a, dc,/dt = -Jl -1J, r= 1 dc,/dt = Jr-l -J,; for r > 1 (5) for the concentrations, c, = [C,], of each size r of cluster, where Note that the total caprylate concentration defined by (7) is a constant (dp/dt = 0),as we would expect.One point of physical significance which emerges from an analysis of the Becker-Doring equations [eqn. (5)] is the concept of metastability. Rigorous mathematical analysis for the case of nucleating vapours shows that there is a certain critical cluster size above which all growth rates remain small for an exponentially long time (the induction period) prior to rapid growth of larger cluster^.'^.^^ This is consistent with experimentally observed nucleation behaviour, and is used in the modelling of nucleation and crystal growth.Note that the assumption of single-particle accretion and fragmentation is a special case of the discrete coagulation-fragmentation equa-tions which can be used to model more general processes, but at considerably more computational expense. '' The Becker-Doring formalism can readily be extended to handle the kinetics of the experiments studied here. To include the effect of ethyl caprylate hydrolysis, both catalysed and uncatalysed, we add the following irreversible steps EC +C; rate = koe, EC + C, 4C + C,; rate = k,ec, for r > rcrit (IV) where e = [EC] and rcrit is some critical cluster size below which catalysis of ethyl caprylate hydrolysis is not possible. This constitutes a minimal modification of the Becker-Doring scheme.The modified Becker-Doring equations are then given by de/dt = -J, dc,/dt = Jr-l -J,; for r > 1 where In this case, the total caprylate concentration defined by m p = [c],,, = e + 1rc, r= 1 is an invariant of the kinetics. If we are to implement this model in a numerical fashion we need some way of estimating the plethora of rate coeffi- cients that arise in these equations. In principle, this can be done through an appeal to equilibrium statistical mechanics and diffusion theory. It is important to recognise that all of these rate constants implicitly contain information on the size (or, equivalently, the surface area) of the micellar clusters. This model, defined by its infinite set of rate equations [eqn. (S)], is mathematically very complicated to study and has yet to be explored fully, but we hope that it will be the subject of future work.However, preliminary numerical calculations provide encouraging evidence that this model does contain the correct features to describe self-replicating micelles. Note also that we can develop a continuum version of this discrete set of ordinary differential equations, which is a partial differential equation in terms of the variables r and t, coupled with an ordinary differential equation for cl. The detailed form of this continuum limit is not as easy to deter- mine as it appears at first sight. However, we know that an integral over the range of cluster sizes must be involved, from the form of the discrete Becker-Doring equation for c,.Such equations can have finite time singularities that can be analysed by asymptotic techniques, as shown in ref. 18. These singularities would correspond to a well defined induction period, similar to that which we analyse in the Appendix for eqn. (4). In other words, there is a good chance that we can construct an asymptotic, metastable solution, of the form shown to exist in ref. 16. Some rigorous analysis along these lines is already in progress.” Note that the more macroscopic chemical kinetic models which we have invoked to account for both ettringite nucle- ation during cement hydrationg and the formation of micelles from monomers in Section 3, most likely emerge as crude mean-field approximations to the mesoscopic theory described in this section.However, even the Becker-Doring equations do not provide a truly microscopic model, as they say nothing about the role of fluctuations on the distribution of cluster sizes; they are concerned only with the average behaviour of such distributions. We thank Pier Luigi Luisi for many stimulating discussions; P.V.C. also acknowledges his hospitality during several visits to E.T.H. We are also indebted to Peter Walde for his help in clarifying many aspects of the experimental work. We are grateful to Jack Carr and Oliver Penrose for sharing their ideas on the application of the Becker-Doring equations to the kinetics of micelle formation, and to Denver Hall for some helpful comments.Appendix Asymptotic Analysis We begin by defining the dimensionless variables x = CECl/~EClO~Y = ccl/cEclo z = [C,]/[ECl0; z = kot (Al) In terms of these variables, eqn. (4) becomes dx _---x -&-1nxz2dz dY -= x -Icnydz _-dz -& -lXZ2 + Icydz subject to the initial conditions x(0) = 1, y(0) = a, z(0) = 0, where a = [C],/[ECl0; E = ko/k2[EC];; u = k,/k, (A3) Here, a is the dimensionless initial concentration of caprylate monomers, E is a measure of the rate of the uncatalysed decomposition of ethyl caprylate relative to the rate of the autocatalytic step, and K is a measure of the rate of the uncatalysed production of micelles from caprylate monomers relative to the rate of the uncatalysed decomposition of ethyl caprylate.Since eqn. (A2) shows that d(x + y + nz)/dz = 0, we can eliminate y and arrive at dx----x -&-lnXz2dz dz ---E -‘xz2 + 1c(1 + a -x -nz)dz We are interested in the case E << 1, when the uncatalysed rate of decay of the ethyl caprylate is slow compared with the rate of the autocatalytic step. We will consider first the problem when a = 0(&li3).A suitable rescaling of the vari- ables is given by z = ,51/3i; x = 1 + &‘/3X = E2/3-.z, a = &‘/3a (A5) J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 At leading order as E +0, eqn. (A4) become subject to X = 0, Z = 0 at i = 0. Clearly, X --i, and hence d2 --j2 + K(ii + 7)di We can now scale K out of this equation by writing 2 = Ic1/3z.7 = K-1/3., a = Ic-1/3~ (A@5 so that dZ--Z2+A+TdT subject to Z = 0 when T = 0.The solution of this simple non-linear equation becomes unbounded, Z + 03, at finite time T = To for all positive values of A. Since Z is a scaled, dimensionless version of the micelle concentration, the time, To, corresponds to the end of the induction period, when the micelle concentration grows rapidly. This is all that we need to know from the asymptotic solution. We can easily calculate the function To(A) numerically by integrating eqn. (A9) and determining when Z becomes large for any given value of A. The function To(A) is illustrated in Fig. Al. Since A is a scaled, dimensionless version of the initial concentration of caprylate monomers, this allows us to determine the duration of the induction period as a function of CClO * We can repeat the above analysis for the case a = 0(1), which is equivalent to A 9 1.We again find that there is a singularity in the leading order solution of the micelle con- centration. However, in this case, we can determine a simple expression for the time at which this singularity occurs, which shows that The curve n/2A1I2 is also shown in Fig. Al, and is clearly in good agreement with the numerically calculated curve To for sufficiently large values of A. If we let to be the duration of the induction period, in terms of the physical variables, 2.0 1.6 Y3 1.2 \ L. 0.8 0.4 t 1 0.0 I I II 0.0 2.5 5.0 7.5 10.0 A Fig. A1 The scaled, dimensionless induction time, To (-) and 7~/2A’’~(---), the asymptote for A 9 1 J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 7r (A 12)to -2(k k, [EC] [C] o) ’2 for k, < k2[EC]i and [Cl0 = O([EC],). Note that T,(O)z 2, so that for ko 3 k,[EC]i and [C], = 0. It is this equation that allows us to determine the value of k, from the given values of to,k,, k, and [EC], in the experiment with [Cl0 = 0. References 1 F. Varela, H. Maturana and R. Uribe, Biosystems, 1974,5,187. 2 P. Bachmann, P. L. Luisi and J. Lang, Nature (London), 1992, 357,57. 3 P. L. Luisi, in Thinking about Biology, ed. W. Stein and F. J. Varela, Addison-Wesley, New York, 1993. 4 A. Malliaris, J. Lang and R.Zana, J. Chem. SOC., Faraday Trans. 1,1986,82, 109.5 S. Colonna, G. Fleischakar and P. L. Luisi, Self-reproduction of Supramolecular Structures, Proc. NATO Advanced Research Workshop Maratea, Italy, 1993, Elsevier, Amsterdam, in the press. 6 P. L. Luisi, personal communication. 7 W. Press, S.Teukolsky, W. Vetterling and B. Flannery, Numeri-cal Recipes, Fortran Version, Cambridge University Press, 1986. 8 J. Billingham and D. J. Needham, Philos. Trans. R. SOC.London, A, 1992, 340, 569; J. Billingham and D. J. Needham, J. Eng. Math., 1993, 27, 113. 9 J. Billingham and P. Coveney, J. Chem. SOC., Faraday Trans., 1993,89,3021. 10 F. Mavelli and P. Walde, unpublished results. 11 J. Carr, 0.Penrose and J. Wattis, unpublished results. 12 Z. Alexandrowicz, J. Phys. A: Math. Gen., 1993, 26, 655; S. 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