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Autocatalyses
Raphaël Plasson∗ , Axel Brandenburg† , Ludovic Jullien‡ , Hugues Bersini§
Abstract
Autocatalysis is a fundamental concept, used in a wide range of domains. From the most general def- inition of autocatalysis, that is a process in which a chemical compound is able to catalyze its own for- mation, several different systems can be described. We detail the different categories of autocatalyses, and compare them on the basis of their mechanistic, kinetic, and dynamic properties. It is shown how autocatalytic patterns can be generated by different systems of chemical reactions. The notion of auto- catalysis covering a large variety of mechanistic re- alisations with very similar behaviors, it is proposed that the key signature of autocatalysis is its kinetic pattern expressed in a mathematical form.
Keywords chemical network, autoinduction, tem- plate, competition, mechanism
Contents
1 Introduction 1
2 Autocatalysis: a Practical Definition 2 2.1 A Kinetic Signature............... 2 2.2 Potential vs Effective Autocatalysis..... 2
3 Mechanistic Distinctions 2 3.1 Template Autocatalysis............. 2 3.2 Network Autocatalysis............. 3 3.3 Autoinductive Autocatalysis.......... 5
4 Embedded Autocatalyses 6 4.1 Dynamical Distinctions............. 6 4.2 Comparative Efficiency of Direct and Au- toinductive Autocatalyses........... 6 4.3 From Autocatalytic Processes towards Au- tocatalytic Sets.................. 7
5 Conclusion 7
6 Appendix 7 6.1 Indirect autocatalysis.............. 7 6.2 Autoinductive autocatalysis.......... 8 6.3 Iwamura’s model................ 9 ∗Keio University, Yokohama, Japan †Nordita, Stockholm University, Stockholm, Sweden ‡ENS, Paris, France §Iridia, ULB, Brussels, Belgium
1 Introduction
The notion of “autocatalysis” was introduced by Ost- wald in 1890^1 for describing reactions showing a rate acceleration as a function of time.^1 It is for example the case of esters hydrolysis, that is at the same time acid catalyzed and producing an organic acid.^2 Defined as a chemical reaction that is catalyzed by its own products, it has quickly been described on the basis of a character- istic differential equation.3, 4^ Typically used to describe complex behaviors of chemical systems, like oscillatory patterns,^5 it has immediately appeared to be essential for the description of biological systems: growth of individual living beings,^6 population evolution^7 or gene evolution.^8 Extending this concept from a chemical description to a more open context was initially carefully described as an analogy, sometime qualified by the more general notion of “autocatakinesis”.9, 10^ However, this eventually leads to an overgeneralization of the term of autocatalysis, tending to be assimilated to the notion of “positive feedback”, for example in economy.^11 The notion of autocatalysis is now actively being used for describing self-organizing systems, namely in the field of emergence of life and artificial life. Autocatalytic pro- cesses are the core of the mechanisms leading to the sym- metry breaking of chemical compounds towards homochi- rality,12, 13^ and could be identified in several experimental systems.14, 15^ However, how such autocatalytic processes shall manifest is still under heavy debate.16, 17 The purpose of this article is thus to clarify the meaning of chemical autocatalysis and this effort will be under- taken by covering these following points:
- What is autocatalysis for a chemical system? On the basis of the general description of a process allow- ing a chemical compound to enhance the rate of its own formation, autocatalysis is defined by a kinetic signature, expressed in a mathematical form.
- How can an autocatalytic process be realized? As many mechanisms can reduce to the same macro- scopic kinetic laws exhibiting autocatalysis, the focus is put on several mechanistic realisations of auto- catalytic processes, based on simple models further illustrated by concrete chemical examples.
- How can autocatalysis be observed and character- ized? The focus is put on the dynamic properties, showing that this observable is the direct conse- quence of the kinetic pattern, rather than the un- derlying mechanism.
arXiv:1006.2634v2 [q-bio.BM] 22 Oct 2010
2 Autocatalysis: a Practical Definition
- What is the role of autocatalysis? Embedded in non- equilibrium reaction network, the competition be- tween autocatalytic processes allows the onset of chemical selection, that is the existence of bifurca- tion phenomena allowing the extinction of some com- pounds in favor of others.
2 Autocatalysis: a Practical
Definition
2.1 A Kinetic Signature
From its origin, the notion of autocatalysis has focused on the kinetic pattern of the chemical evolution.^3 The general definition of autocatalysis as a chemical process in which one of the products catalyzes its own formation can be mathematically generalized as:
d xi d t
= k ( X ) · x ni + f ( X ), k > 0; n > 0; | k | | f | (1)
X is the vector of all the concentrations x (^) j. An autocatal- ysis for the compound xi exists when the conditions of eq. (1) are fulfilled. The term k ( X ) · x ni describes the auto- catalytic process itself, while f ( X ) describes the sum of all other contributions coming from the rest of the chemical system. We have an effective practical definition of the con- cept of autocatalysis, based on a precise mathematical formulation. The causes of this kinetic signature can be investigated, searching what mechanism is responsible for the autocatalytic term. This leads to the discovery of a se- ries of different kinds of autocatalysis processes, and their respective effect, describing what observable behavior is generated by the autocatalytic term (see fig. 1).
2.2 Potential vs Effective Autocatalysis
This kinetic definition is purely structural. As a mat- ter of fact, a system may contain potential autocatalysis i.e. an autocatalytic core exists in the reaction network. However, in the absence of some specific conditions nec- essary for this autocatalysis to be effective , the potential autocatalysis may be hidden by other kinetic effects, and thus not manifests its behavior in practice. Possibly, in eq. (1), the term f ( X ) may simply over- whelm the autocatalytic process. This is typically the case when an autocatalysis is present together with the non-catalyzed version of the same reaction, that may not be negligible in all conditions. A simple example is a system simultaneously containing a direct autocatalysis A + B −→ 2 B , concurrent with the non autocatalytic reac- tion A −→ B. The autocatalytic process follows a bimolec- ular kinetics, and will be more efficient in a concentrated than in a diluted solution. The dynamic profile of the re- action is thus sigmoidal for high initial concentration of A , but no more for low initial concentration (see fig. 2(a-b)). It can also be seen that the term k ( X ) may vary during the reaction process. In a simple autocatalytic process as described above, k is proportional to the concentration in
A , and is thus more important at the beginning of the re- action (thus an initial exponential increase of the product B ) that at the end (thus a damping of the autocatalysis) resulting in a global sigmoidal evolution. In systems were the influence of A on k is weaker, as detailed further, an undamped autocatalysis will be observed characterized by an exponential variation until the very end (see fig. 2(c)).
3 Mechanistic Distinctions
How can this kinetic pattern be realized? Let us now de- tail several types of mechanisms. They can all be reduced, in some conditions, to the autocatalysis kinetic pattern of eq. (1). All of them will be equally defined in the paper as autocatalytic, while this status may have been disputed in the past on account of the distinct chemical realisations. In the following, we emphasize the major mechanistic pattern to eventually be reduced to an equivalent kinetic autocatalysis, and discuss where their difference comes from.
3.1 Template Autocatalysis
The simplest autocatalysis is obtained by the X −→ 2 X pattern. It can be represented by:
A + B
k 1 − )* − k − 1
B + B (2)
The corresponding network is given in fig. 3(a). It can further be decomposed through the introduction of an intermediate compound C :
A + B
Γ 1 − )* − C (3)
C
Γ 2 − )* − B + B (4)
The corresponding network is given in fig. 3(b). The first mechanism entails the following kinetic evolu- tion: d b d t
d a d t
= k 1 ab − k − 1 b^2 (6)
This can be expressed as a chemical flux ϕ = d d^ bt , by relying on the Mikulecky formalism:18–
ϕ = Γ 1 ( VAVB − V (^) B^2 ) (7)
VA =
a KA
VB =
b KB
Γ 1 = k 1 · KAKB = k − 1 · K B^2 (10)
k 1 and k − 1 are the kinetic constant rates of the reaction 1 in the direct and reverse direction. KA and KB are the thermodynamic constant of formation of compounds A and B. Formally there is a linear flux ϕ of transformation of A into B , coupled to a circular flux of same intensity from B
3 Mechanistic Distinctions
(a) Direct (b) Direct with intermediate
(c) Indirect (d) Autoinductive
(e) Iwamura et al^25 system
A₁
B₁
A₂ B₂
A₃
B₃
φ₁ φ₁ B₄ A₄
φ₂
φ₂
φ₃ φ₃
φ₄
φ₄
(f) Collective
Figure 3 – Reaction network of different autocatalytic pro- cesses of spontaneous transformation of A into B (a-d), of A + X into AX (e), and of Ai into Bi (f). The indicated fluxes correspond to what is observed within the QSSA.
3.2.1 Indirect Autocatalysis:
The autocatalytic effect can be indirect when reactant and products never directly interact:
A + D
Γ 1 − )* − C (16)
C
Γ 2 − )* − B + E (17)
E
Γ 3 − )* − B (18)
B
Γ 4 − )* − D (19)
There is no direct A/B coupling, nor direct 2 B formation, but the presence of a dimeric compound C. The network decomposition of this system (see fig. 3(c)) implies once again a linear flux of transformation of A into B , linked to a large cycle of reaction transforming B back to B. This system is still reducible to an X −→ 2 X pattern. The QSSA for compounds C , D , E allows to express the reaction flux as:
ϕ =
1 Γ 1 +^
1 Γ 2 +^
VA Γ 4 +^
VB Γ 3
Ä
VAVB − V B^2
ä (20)
The details of the calculations are given in appendix. When the terms VA/ Γ 4 and VB / Γ 3 are small compared to either Γ− 1 1 or Γ− 2 1 (i.e. when at least one of the two reactions of eq. (16)-eq. (17)) is kinetically limiting), the
(a) Direct (b) Indirect
(c) Autoinductive (d) Collective
Figure 4 – Time evolution of compound concentrations for different autocatalytic processes of spontaneous transforma- tion of A into B ( KA = 1 M and KB = 100 M) in a logarithmic scale for concentrations (a-c), or logarithmic scales for both time and concentrations (d). K and concentrations are in M, times in s, and Γ in M.s −^1_. (a): fig. 3(b),_ Γ 1 = 1 , Γ 2 = 10 −^4 , KC = 0. 01 ; (b): fig. 3(c), Γ 1 = Γ 2 = Γ 3 = Γ 4 = 10 (except the values indicated on the graph), KC = KD = KE = 0. 01 ; (c): fig. 3(d), Γ 2 = Γ 3 = 100 , KC = KE = 1 , KE ∗ = 10 ; (d): fig. 3(f), Γ 1 = 100 , Γ 2 = 1_._
system behaves like a simple autocatalytic system, with ϕ ∝ a · b before the reaction completion, with a progres- sive damping of the exponential growth as long as A is consumed. When the term VA/ Γ 4 is predominant (i.e when the reaction of eq. (19) is kinetically limiting), the flux is ϕ ∝ b : the profile remains exponential up to the reaction completion, with no damping due to A consump- tion. When the term VB / Γ 3 is predominant (i.e when the reaction of eq. (18) is kinetically limiting), the flux is ϕ ∝ a : the autocatalytic effect is lost (see fig. 4(b)). Network autocatalysis is probably the most common kind of mechanisms. A typical biochemical example is the presence of autocatalysis in glycolysis.26, 27^ In this system, there is a net balance following the X −→ 2 X pattern. ATP must be consumed to initiate the degradation of glucose, but much more molecules of ATP are produced during the whole process. While these systems are effectively autocatalytic, there is obviously no possible “templating” effect of one molecule of ATP to generate another one.
3.2.2 Collective Autocatalysis:
More general systems, reminiscent of the Eigen’s hy- percycles,^28 are responsible of even more indirect auto- catalysis. No compound influences its own formation rate, but rather influences the formation of other compounds, which in turn influence other reactions, in such a way that the whole set of compounds collectively catalyzes its own formation. A simple framework can be built from the association of several systems of transformation Ai −→ Bi , each Bi
3 Mechanistic Distinctions
catalyzing the next reaction (see fig. 3(f)):
Ai + Bi − 1
Γ i − )* − Bi + Bi − 1 (21) (with i = {1, 2, 3, 4} and B 4 ≡ B 0 )
There are four independent systems, only connected by catalytic activities. If the system is totally symmetric, then all bi are equal, and all ai are equal, so that the rates become:
ϕi = Γ i VBi − 1 ( VAi − VBi ) (22) ϕ = Γ VB ( VA − VB ) (23)
This leads to a collective autocatalysis with all compounds present. They mutually favor their formation, which re- sults in an exponential growth of each compound (see fig. 4(d) dotted curve). With symmetrical initial conditions (i.e. identical for the four systems), the system strictly behaves autocatalytically. If the symmetry is broken, e.g. by seeding only one of the Bi , the system acts with delays. The evolution laws are sub-exponential, of increasing order; at the very beginning of the reaction, considering that Ai do not significantly change and that Bi are in low concentrations, we obtain ϕi ∝ ti −^1. Seeding with B 1 , the compound B 2 evolves in t^2. Its impact on compound B 3 induces an evolution in t^3. In its turn, the impact of compound B 3 on compound B 4 induces an evolution in t^4. The compound 1 at first remains constant, and it is only following a given delay that it gets catalyzed by B 4 (see fig. 4(d)). This system is actually not characterized by a direct cyclic flux, but by a cycle of fluxes influencing each other and resulting in a cooperative collective effect:
( A 1 + A 2 + A 3 + A 4 ) + ( B 1 + B 2 + B 3 + B 4 ) (24) −→ 2 ( B 1 + B 2 + B 3 + B 4 ) (25)
The simultaneous presence of all different compounds is needed to observe a first order autocatalytic effect. Given asymmetric initial conditions, a transitory evolution of lower order is first observed, until the formation of the full set of compounds. A typical example of collective autocatalysis is observed for the replication of viroids.^29 Each opposite strand of cyclic RNAs can catalyze the formation of the other one, leading to the global growth of the viroid RNA in the infected cell.
3.2.3 Template vs Network Autocatalysis:
All the preceding systems can be reduced to a X −→ 2 X pattern. This is characterized by a linear flux of chemical transformations, coupled to an internal loop flux: for each molecule (or set of molecules) A transformed into B , one B is transformed and goes back to B , following a more or less complex pathways. They can be considered as mech- anistically equivalent: a seemingly direct autocatalysis may really be an indirect autocatalysis once its precise mechanism is known, decomposing the global reaction into several elementary reactions. Practically, autocatalysis will be considered to be direct (or template) when a dimeric complex of the product is
formed (i.e. allowing the “imprint” of the product onto the reactant). If such template complex is never formed, we preferentially speak of network autocatalysis, in which the X −→ 2 X pattern only results from the reaction balance.
3.3 Autoinductive Autocatalysis
Some reactions are not characterized by a X −→ 2 X pattern, but still exhibit a mechanism for the enhancement of the reaction rate by the products. This is typically the case for systems where the products increase the reactivity of the reaction catalyst rather than directly influencing their reaction production itself. These systems still possess the kinetic signature of eq. (1), but are sometime referred as “autoinductive” instead of “autocatalytic”.^17
3.3.1 Simple network:
Let us take a simple reaction network of a transforma- tion A −→ B catalyzed by a compound that can exist under two forms E/E ∗, E ∗^ being the more stable one. These two forms of the catalyst interact differently with the product B (see fig. 3(d)):
A + E
Γ 1 − )* − C (26)
C
Γ 2 − )* − B + E (27)
C
Γ 3 − )* − B + E ∗^ (28)
There is no dimeric compound in the system, even indi- rectly formed. Provided the catalyst, present in C , E , and E ∗, is in low total concentration, the QSSA implies the presence of two fluxes: the transformation of A into B catalyzed by E of intensity ϕ , and the transformation of E ∗^ into E catalyzed by B of intensity " , with ϕ ". Assuming that E ∗^ is very stable compared to E and C , this decomposition gives (see appendix for details):
ϕ =
Γ 1 Γ 2 V E^0 ∗
Γ 1 VA + Γ 2 VB
( VB VA − V B^2 ) (29)
The autoinduction is kinetically equivalent to the indi- rect autocatalysis mechanism:
- When Γ 2 Γ 1^ K KB A , the flux ϕ is Γ 1 V E^0 ∗ ( VA − VB ): the system is non-autocatalytic.
- When Γ 2 ≈ Γ 1^ K KB A , the flux ϕ is Γ 2
V (^) E^0 ∗ V A^0 ( VAVB^ −^ V^
2 B ): the system is simply autocatalytic.
- When Γ 2 Γ 1^ K KB A , the flux ϕ is Γ 2 V E^0 ∗
Å
VB −
V (^) B^2 VA
ã : the system presents an undamped autocatalysis.
Following the kinetic analysis, the behavior is similar to the time evolution of autocatalytic systems (See fig. 4(c)). The behavioral equivalence of these two systems (kinet- ically equivalent but mechanistically very different) will be investigated in more details in the next section.
5 Conclusion
(a) Sharp bifurcation depending on the relative values of α and β for moderate reactivities.
(b) Different zones of behaviors: majority of A for α , β 1 , ma- jority of B 1 for α > β, majority of B 2 for α < β, and coexistence of B 1 and B 2 for α , β 1_._
Figure 5 – Competition between template and autoinductive autocatalysis, generating respectively B 1 and B 2 compounds from the same A compound. Incoming flux of A, and outgoing fluxes of B 1 and B 2 , 10 −^5 M.s −^1_. KA_ = 1 , KB 1 = KB 2 = 100_. Direct autocatalysis:_ Γ AC = 10 −^2 · α, Γ N C = 10 −^6 · α. Autoinduction, according to fig. 3(d): Γ 1 = β, Γ 2 = Γ 3 = 100 · β, KC = KE = 1 ; KE ∗ = 10_._
there is a flush of the system, and neither B 1 nor B 2 can be maintained. For fast kinetics, the system is close to equilibrium, the compounds B 1 and B 2 being both present in proportion to their respective stability (see fig. 5(b)). Such result is well known for open flow Frank systems.^39
4.3 From Autocatalytic Processes
towards Autocatalytic Sets
These competitive systems are able to dynamically maintain a set of components, to the detriment of others. These autocatalytic networks must however not be con- fused with autocatalytic sets. This latter notion is rather popular in the artificial life literature, but relies much more on the cooperation between autocatalytic mecha- nisms than on the competition that has just been detailed here. This implies a notion of material closure of the
system and of self maintenance of the whole network by crossing energetical fluxes.40–42^ Confusion among these different phenomena can be pinpointed in the literature,^17 when the failure of autoinductive sets to be maintained do not originate from a difference of behavior between autocatalytic and autoinductive mechanisms, but from a defect in the closure of the system (e.g. induced by the leakage of some components).
5 Conclusion
Important distinctions need to be made between mech- anistic and dynamic aspects of autocatalysis. One single mechanism can produce different dynamics, while iden- tical dynamics can originate from different mechanisms. Thus, a pragmatic definition of autocatalysis have to be based on a kinetic signature, in order to classify the sys- tems according their observable behavior, rather than on a mechanistic signature, that would instead classify the systems according to the origin of their behavior. All the different autocatalytic processes described in this work are able to generate autocatalytic kinetics. They can con- stitute a pathway towards the onset of “self-sustaining au- tocatalytic sets”, as chemical attractor in non-equilibrium networks. However, the problem of the evolvability of such systems must be kept in mind.^43 If a system evolves towards a stable attractor, no evolution turns out to be possible. There is the necessity of “open-ended” evolu- tion^44 i.e. the possibility for a dynamic set to not only maintain itself (i.e. as a strict autocatalytic system) but also to act as a “general autocatalytic set”, redounding upon the concept originally introduced by Muller^8 for the autocatalytic power linked to mutability of genes. For example, insights can be gained by a deeper and renewed study of the evolution of prions as a simple mechanism of mutable autocatalytic systems.^45
6 Appendix
The kinetic behavior of three different mechanisms for autocatalytic transformations have been studied in details. The methodology consists in establishing the different chemical fluxes of the network. The relationship between these fluxes can be simplified by assuming the QSSA for relevant compounds. The purpose is then to establish the expression of the transformation flux ϕ as a function of the concentration of the reactants and the products.
6.1 Indirect autocatalysis
The four fluxes of fig. 3(c) can be written as:
ϕ 1 = Γ 1 ( VAVD − VC ) (37) ϕ 2 = Γ 2 ( VC − VB VE ) (38) ϕ 3 = Γ 3 ( VE − VB ) (39) ϕ 4 = Γ 4 ( VB − VD ) (40)
6 Appendix
The QSSA for D comes down to ϕ 1 ' ϕ 4 :
Γ 1 VAVD − Γ 1 VC = Γ 4 VB − Γ 4 VD (41)
(Γ 1 VA + Γ 4 ) VD = Γ 4 VB + Γ 1 VC (42)
VD =
Γ 4 VB + Γ 1 VC
Γ 1 VA + Γ 4
Replacing VD by eq. (43) in eq. (37) gives:
ϕ 1 = Γ 1
Ç
VA
Γ 4 VB + Γ 1 VC
Γ 1 VA + Γ 4
− VC )
å (44)
Γ 1 VA + Γ 4
VAVB + Γ 1 VC
Γ 1 VA + Γ 4
Γ 1 VA + Γ 4
( VAVB − VC ) (46)
The QSSA for E comes down to ϕ 2 ' ϕ 3 :
Γ 2 VC − Γ 2 VB VE = Γ 3 VE − Γ 3 VB (47)
VE =
Γ 2 VC + Γ 3 VB
Γ 3 + Γ 2 VB
Replacing VE by eq. (48) in eq. (38) by Eq.gives:
ϕ 2 = Γ 2
Ç
VC − VB
Γ 2 VC + Γ 3 VB
Γ 3 + Γ 2 VB
å (49)
= Γ 2 VC
Γ 3 + Γ 2 VB
− Γ 2 VB
Γ 3 VB
Γ 3 + Γ 2 VB
Γ 3 + Γ 2 VB
( VC − V B^2 ) (51)
At last, the QSSA for C comes down to ϕ 1 ' ϕ 2 = ϕ. Combining eq. (46) and Eq, eq. (51) gives:
VC =
Γ′ 1 VAVB + Γ′ 2 V B^2
with Γ′ 1 =
Γ 1 VA + Γ 4
and Γ′ 2 =
Γ 3 + Γ 2 VB
Replacing VC by eq. (52) in eq. (46) gives:
ϕ = Γ′ 1 VAVB − Γ′ 1
Γ′ 1 VAVB + Γ′ 2 V B^2
VAVB −
V B^2 (56)
( VAVB − V B^2 ) (57)
Replacing Γ′ 1 and Γ′ 2 by their expression given in eq. (53) and eq. (54) then gives:
ϕ =
VAVB − V B^2
1 Γ 1 +^
1 Γ 2 +^
VA Γ 4 +^
VB Γ 3
6.2 Autoinductive autocatalysis
The three fluxes of fig. 3(d) are:
ϕ 1 = Γ 1 ( VAVE − VC ) (59) ϕ 2 = Γ 2 ( VC − VB VE ) (60) ϕ 3 = Γ 3 ( VB VE ∗ − VC ) (61)
The QSSA for C comes down to ϕ 1 + ϕ 3 ' ϕ 2 , and the QSSA for E comes down to ϕ 1 ' ϕ 2. This implies that ϕ 3 ϕ 1 , so that with ϕ 3 = " and, ϕ 1 = ϕ , we obtain:
ϕ 2 = ϕ + " ' ϕ (62)
In that context, eq. (61) gives:
VC = VB VE ∗ −
Combining eq. (59), eq. (60) in eq. (62) then gives:
Γ 2 VC − Γ 2 VB VE = Γ 1 VAVE − Γ 1 VC + " (64)
VE =
(Γ 1 + Γ 2 ) VC − "
Γ 1 VA + Γ 2 VB
Replacing VC by its value given in eq. (63) leads to:
VE =
(Γ 1 + Γ 2 ) VE ∗^ VB −
Γ 1 +Γ 2 +Γ 3 Γ 3^ " Γ 1 VA + Γ 2 VB
VE '
(Γ 1 + Γ 2 ) VE ∗ VB
Γ 1 VA + Γ 2 VB
The flux of destruction of A can be computed by replacing VE in eq. (59) by eq. (67) (computing the flux of formation of B from eq. (60) would of course give the same result):
ϕ = Γ 1
Ç
VA
(Γ 1 + Γ 2 ) VE ∗^ VB
Γ 1 VA + Γ 2 VB
− VB VE ∗
å (68)
(Γ 1 + Γ 2 ) VE ∗ VB VA − VB VE ∗ (Γ 1 VA + Γ 2 VB )
Γ 1 VA + Γ 2 VB
= Γ 1 Γ 2 VE ∗
VB VA − V B^2
Γ 1 VA + Γ 2 VB
The law of conservation of E compounds leads to:
ϕ =
Γ 1 Γ 2 V E^0 ∗ ( VB VA − V B^2 )
(Γ 1 VA + Γ 2 VB )( 1 + rC VC + rE VE )
with rC = KC /KE ∗ and rE = KE /KE ∗. Assuming that E ∗^ is the much more stable than C and E , rC and rE 1, so that we finally obtaina:
ϕ =
Γ 1 Γ 2 V E^0 ∗
Γ 1 VA + Γ 2 VB
( VB VA − V B^2 ) (72)
aWithout the hypothesis of a large stability of E ∗, not neglecting the rC and rE terms eventually leads to add VB terms to the denominator, which will tend to destroy the autocatalytic effect.
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