Complex Autocatalysis
in Simple Chemistries
Abstract Life on Earth must originally have arisen from abiotic
chemistry. Since the details of this chemistry are unknown, we wish to
understand, in general, which types of chemistry can lead to complex,
lifelike behavior. Here we show that even very simple chemistries
in the thermodynamically reversible regime can self-organize to form
complex autocatalytic cycles, with the catalytic effects emerging from
the network structure. We demonstrate this with a very simple
but thermodynamically reasonable artificial chemistry model. By
suppressing the direct reaction from reactants to products, we obtain
the simplest kind of autocatalytic cycle, resulting in exponential
growth. When these simple first-order cycles are prevented from
forming, the system achieves superexponential growth through more
complex, higher-order autocatalytic cycles. This leads to nonlinear
phenomena such as oscillations and bistability, the latter of which
is of particular interest regarding the origins of life.
Nathaniel Virgo*,**
University of Tokyo
and
Tokyo Institute of Technology
†
Takashi Ikegami
University of Tokyo
Simon McGregor
University of Sussex
‡
Keywords
Autocatalysis, origin of life,
thermodynamics, chemical kinetics,
self-organization
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1 Introduction
How can abiotic chemistry give rise to phenomena such as replication, metabolism, and evolution?
This question is at the core of the origins of life as a field of inquiry, and there are many approaches
to it. One such methodology is artificial chemistry, the study of more-or-less abstract artificial chemical
reaction networks and their emergent dynamical properties. This field can be traced back to the
work of Fontana and Buss [5] and Kauffman [6]. More recently it has been used to show that evo-
lution can occur without the need for template replication [12], and to investigate the mechanisms
behind real reactions, particularly those that are too complex to model with traditional approaches,
such as as HCN polymerization [1].
In these models, thermodynamic principles play at best a minor role. However, the nature of
living organisms as far-from-equilibrium structures is fundamental to our understanding of biology
at the chemical level. Our work aims to show that new insights can be gained by putting thermo-
dynamics at the heart of artificial chemistry approaches.
Recently we presented a simple yet thermodynamically reasonable artificial chemistry [15]. When
held out of equilibrium, this chemistry formed moderately large and complex autocatalytic cycles
* Contact author.
** Ikegami Laboratory, University of Tokyo, Japan, and Earth-Life Science Institute, Tokyo Institute of Technology, Japan. E-mail:
nathanielvirgo@gmail.com
† Ikegami Laboratory, University of Tokyo, Japan. E-mail: ikeg@sacral.c.u-tokyo.ac.jp
‡ University of Sussex, U.K. E-mail: s.mcgregor@sussex.ac.uk
© 2016 Massachusetts Institute of Technology Artificial Life 22: 138–152 (2016) doi:10.1162/ARTL_a_00195
N. Virgo et al.
Complex Autocatalysis in Simple Chemistries
(that is, moderately large sets of species that are collectively capable of exponential growth). Unlike
many previous studies, this chemistry did not include any a priori notion of catalysis. Instead, cat-
alytic effects emerged as dynamical properties of the reaction network. The system was allowed to
dynamically “choose” the direction of its reactions according to thermodynamic principles, and it
was this property that allowed it to self-organize into autocatalytic cycles.
This phenomenon of self-organizing autocatalysis happens under some circumstances but not
others. In order to fully understand the result, we must investigate which classes of reaction net-
works will lead to autocatalysis and which will not. Moreover, the resulting cycles can be more or less
complex, and we wish to understand what factors affect this. A comprehensive answer to this ques-
tion would be a great help in understanding the origins of life. If we know which types of chemical
system are likely to lead to complex, metabolism-like reactions, then it will give us clues about what
to look for in identifying the types of abiotic chemistry from which life eventually arose.
In this article we continue this project. By constructing a series of simple models, we first dem-
onstrate that autocatalysis arises quite easily in a fairly simple class of chemistries. More importantly,
we show that it must necessarily arise in any chemistry that meets a fairly mild set of preconditions. We
demonstrate this by blocking the simplest form of autocatalysis in our models, and observing that
autocatalysis still occurs, but this time through a more complicated, second-order mechanism.
As with all nonequilibrium phenomena, the autocatalysis in our systems is caused by the tendency of
energy (and other conserved currents) to “flow downhill” from more to less constrained states, that
is, from low to high entropy. Informally, it seems that nature likes to take the “path of least resistance”
in order to achieve this. If the easiest path is a simple, direct chemical reaction, then no complex be-
havior will be observed. Without a direct pathway from initial conditions to equilibrium, autocatalytic
pathways tend to be observed. Blocking the simplest type of autocatalytic cycle results in the formation
of more complex cycles, giving rise to nonlinear phenomena such as bistability and oscillations.
Whether the easiest path toward equilibrium is given by a direct reaction, or a simple or more
complex autocatalytic cycle, is determined by the topology of the reaction network. In this article
we start with a simple polymerization chemistry and block reaction pathways simply by saying that
certain molecules cannot exist. In real chemical systems the network topology depends on the details
of molecular physics, which determines the set of molecules that can exist and the reactions that
can occur between them.
Autocatalytic cycles can only emerge if they are possible within the reaction network. In this
article we deal with reversible chemistries, in which the direction in which reactions occur is not
fixed a priori. In this case we will see that even the simplest reaction schemes contain many auto-
catalytic subnetworks.
Briefly, our argument is that for a non-driven thermodynamic chemical system in the reversible
regime, there exists a unique stable fixed point, namely the thermodynamic equilibrium. However (as
we demonstrate), it is quite possible for there to be other fixed points, which must necessarily be
unstable. At these nonequilibrium fixed points, some of the speciesʼ concentrations are zero. For
trajectories in a region surrounding such a fixed point these concentrations are small but nonzero,
and must increase at an increasing rate as the trajectory accelerates away from the repelling fixed point.
The results we present are very robust. With one minor exception (the oscillatory behavior shown
in Figure 3), they require no parameter tuning at all. Moreover, our mathematical arguments are
applicable to a much broader class of chemical systems than the specific models we present. Thus,
although our model is relatively simple and abstract, we expect that in future work similar results will
be found in much more realistic scenarios.
2 Thermodynamics in Artificial Chemistry
A glance at a chemistry textbook will reveal a large number of formulae that appear to be largely
empirical in nature, and this can give the impression that chemical thermodynamics is something very
contingent upon the details of molecular interactions. Perhaps because of this, there is a tendency
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in artificial chemistry to “abstract away” thermodynamics, choosing a set of reactions arbitrarily,
without regard to energetic considerations. This may be justified by saying that some of the re-
actions are driven by an energy source that is external to the system and is not explicitly modeled. This
methodology dates back to the origins of artificial chemistry in the work of Fontana and Buss [5].
Such an abstraction is of course perfectly valid, but we hope to show that important insights can
be gained if we choose not to make it. The dynamics of a system where there are only a few energy
sources and the majority of reactions must “flow downhill” are quite different from those in which
the reactions proceed in arbitrarily chosen directions. Moreover, the flow of free energy through the
network can be tracked, giving us insights into its dynamics that would be missing if the energetics
were not modeled.
Although chemical thermodynamics might appear to be a rather empirical set of results, its
core principle (the second law) is an extremely fundamental property of physical systems. Essentially,
it boils down to a requirement that for a system with no external supply of energy, there must exist
a global Lyapunov function, which may be called the entropy or the free energy, depending on
whether the system is connected to a heat bath. The existence of such a function is derived from
statistical mechanics, and is a consequence of the time reversibility of physics on the microscopic
level. The details of this formalism are beyond the scope of this article and may be found, for
example, in [9]. We mention it in order to emphasize that our results are less contingent upon
the details of molecular interactions than they might at first appear.
The key feature of a thermodynamically reasonable artificial chemistry is that the reaction
network doesnʼt specify the direction of the reactions. If the network includes a reaction, such as
A + B → C, it must also include the reverse reaction, C → A + B. In general, these two
reactions may proceed at different rates. We refer to the difference in their rates as the net rate of
the bidirectional reaction, A + B ⇌ C.
The dynamics must be such that, in the absence of any energy sources, the system will eventually
reach a state of detailed balance in which the net rate of every reaction is zero. That is, every forward
reaction must be balanced, pairwise, by its corresponding reverse reaction. (This point is of fun-
damental importance. In a general artificial chemistry a dynamical equilibrium can be formed by
a cycle, such as A → B; B → C; C → A. In a thermodynamic system with no driven
reactions this is a physical impossibility.) Detailed balance is a fundamental physical property of
thermodynamically closed or isolated chemical systems at equilibrium. It need not apply to a non-
isolated system that has some externally imposed flow of matter or energy, and in particular it
doesnʼt hold in biological systems. Nevertheless, its importance cannot be overstated, because the con-
straints it imposes on the dynamics are felt even when the system is far from equilibrium.
The condition of detailed balance may be seen as a property of time symmetry held by the equilibrium
state. Away from equilibrium this symmetry may be broken. Every net reaction must always proceed in the
direction that takes the system closer to this equilibrium state. Thus, no reaction can occur at a nonzero
net rate unless the system either is in a transient state or is being driven by an external power source.
In many real reactions the equilibrium ratio between products and reactants is so high that the
reverse reaction is never observed in practice; in this case we think of reactions as occurring strictly
in the “downhill” direction. In this article we are concerned with reversible chemistries, in which this
is not the case for any reaction. The chemistry of the formose cycle is a real-world example of such a
reversible network, in which autocatalysis does indeed emerge.
Many of the reactions in a living cell are reversible, but there are also many key irreversible steps.
We strongly suspect that the main results of this article can be extended to chemistries with irre-
versible reactions under some appropriate set of additional assumptions; however, this remains a
task for future work.
In our model we use a simple implementation of chemical dynamics, known as mass-action kinetics.
Essentially this corresponds to a model of a well-mixed reactor in which molecules collide with each
other at random. When two molecules collide, they react with a probability determined by the rate
constant of the reaction. Thermodynamics puts constraints on the values of the rate constants; these
constraints are satisfied by the dynamics described below.
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3 The A-Polymerization Model
When implemented in a thermodynamically plausible way, even very simple reaction schemes can
give rise to interesting behavior. Our model consists of unary strings A, AA, AAA, . . ., which we
write as A1, A2, A3, . . .. In principle these are countably infinite, but we impose a limit for com-
putational reasons. These species can be thought of as (non-oriented) polymers, with A1 (or simply
A) representing a monomer, and Ai a polymer (or oligomer ) of length i. All reactions are of the form
Ai þ Aj⇌Aiþj :
(1)
That is, two oligomers may join together into a longer one, or an oligomer may split into two smaller
ones. Reactions of this form conserve the total number of A monomers.
If every reaction of the form in Equation 1 is included in the network, then the results are not
especially interesting. However, we can construct A-polymerization chemistries with more complex
dynamics by removing particular reactions from the network. In real chemistry, reactions may be
blocked (or happen at very slow rates) for a variety of reasons, such as conservation laws or steric
effects. Our goal here is not to model such effects specifically, but rather to understand, in general,
the effects of imposing constraints on the reaction network. We write R for the set of reactions
included in the network, and we call the members of R permitted reactions.
→ Ai +j and the decomposition reaction Ai+j
⇌ Ai +j may be thought of as two reactions occurring simultaneously: the
A reaction Ai + Aj
→ Ai + Aj . Writing ak for
synthesis reaction Ai + Aj
the concentration of species Ak, mass-action kinetics tells us that the synthesis reaction occurs at a
rate ksaiaj and the decomposition reaction occurs at a rate kdaij, where the constants ks and kd are
known as the rate constants for the two reactions. This means that the reaction uses up Ai and Aj and
generates Aij at a (possibly negative) net rate given by ksaiaj − kdaij, which we denote RAi þAj⇌Aiþj .
The constants ks and kd cannot be chosen arbitrarily, but instead are constrained by thermo-
dynamics. In general they can be different for different reactions, but in our models we take ks
and kd to be the same for every reaction, with one small exception, as explained below. (In reality,
reaction rates can vary between reactions by many orders of magnitude, but we have made this
assumption for the sake of simplicity. The key results of this article do not depend strongly upon
this assumption.)
In general, a set of mass-action equations may be thought of as a meanfield approximation to a
stochastic model. In our case we may think of the underlying stochastic picture as follows: There is
a well-mixed container containing a solution of oligomers of various lengths. In order for two oligomers
to join together, their ends must meet, and since every oligomer has two ends, we can assume the
probability of this happening is proportional to the product of the two speciesʼ concentrations, regard-
less of the moleculesʼ length. (This assumption is standard in the study of polymerization kinetics.) If
the ends of two molecules of species Ai and Aj do meet, there are two possibilities. If the reaction Ai +
Aj ⇌ Ai+j 2 R, then a bond is formed at the point where the two ends meet, and a new molecule of
species Ai+j is created. Otherwise the reaction is not permitted, and nothing happens; the two molecules
simply bounce off one another unchanged.
In addition to this process, a molecule may spontaneously split into two smaller molecules. An
oligomer of species Ak consists of k A-monomers, linked together by k − 1 bonds. Each of these
bonds has a constant probability per unit time of becoming unstable. If the ith bond of an Ak olig-
omer becomes unstable (with i < k), it will break if the reaction Ai + Ak−i ⇌ Ak 2 R, forming a
molecule each of Ai and Ak−i. Otherwise, nothing happens, and the molecule remains unchanged.
In converting this to a set of differential equations, some care must be taken over reactions Ai +
⇌ Ai+j for which i = j. In the stochastic picture, a molecule of A4 may split into A1 and A3 in
Aj
two different ways: by breaking at its first bond or its third one (assuming such reactions are per-
mitted). In contrast, there is only one way in which it can split into two molecules of A2: by breaking
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Complex Autocatalysis in Simple Chemistries
at its midpoint. Thus, reactions of the form 2Ai ⇌ A2i should happen at half the rate of the other
reactions.
However, as a notational convenience we can get around this by considering Ai + Aj
⇌ Ai+j
⇌ Ai+j to be two distinct reactions, which both happen to have the same set of
and Aj + Ai
products and reactants. Such pairs of reactions are always either both included in R or both excluded.
With these conventions we may write ri, j = 1 if the reaction Ai + Aj ⇌ Ai + j 2 R, and 0 otherwise.
Mass action kinetics then leads to the ordinary differential equations
dai
dt
¼
Xi−1
(cid:2)
ri; j−i ksaj ai−j − kdai
(cid:3)
j¼1
X∞
(cid:2)
þ
ri; j þ rj;i
j¼1
(cid:3)
(cid:2)
kdai þj
− ksai aj
(cid:3)
þ f
i
:
(2)
The term ksajai −j represents the formation of Ai through the joining of Aj to Aj −i; the term kdai rep-
resents the loss of Ai through splitting into Aj and Aj −i; the term kdai +j represents the gain in Ai due to
Ai+j splitting into Ai and Aj; and ksaiaj represents the loss of Ai by joining to Aj to make Ai+j.
i terms represent the flux of each species Ai in or out of the system. These terms may be
set to nonzero values in order to model an open system with external energy sources. In most of this
article these terms will be set to zero, representing a closed system whose free energy comes purely
from a nonequilibrium initial state.
The f
Finally, we must show that these equations are thermodynamically reasonable. For this to be the
case, we must be able to associate with every species Ak a constant number ck, known as the Planck
potential, such that, for every reaction Ai + Aj ⇌ Ai+j ,
Dc;
¼ e
ks
kd
(3)
− c
where Dc is the difference between the total Planck potential of the reactants and the products,
given in our case by Dc = ci + cj − ci+j. It can be shown that if Equation 3 holds, then the system
has a Lyapunov function given by (cid:2)k xk(ck + log xk), which corresponds to the Gibbs free energy
in chemistry. To show that this is the case for our model, we note that Equation 3 tells us that c
i +
c
k = B + Ck for any constants B and C, with
j
Dc being given by B.
i+j is independent of i and j. This is satisfied by c
The notation in terms of ck is a nonstandard one that we have introduced in order to emphasize
the simplicity of the formalism. In chemistry we would more usually talk about Gibbs energies of
formation rather than Planck potentials, and in the standard notation our ck would be written
fGk°/kBT, where kB is Boltzmannʼs constant and T the temperature. Equation 3 usually appears
D
= exp(D
in chemistry textbooks in the form ks/kd
rG °/kBT ). A comprehensive introduction to
chemical thermodynamics, and the language used by chemists to describe it, may be found in [9].
In terms of Gibbs energies, we may further justify our use of constant ks and kd by assuming that
fGA° and that forming each A–A bond
f GA° + (i − 1)EAA. This
fGA° term canceling out because reactions neither
each monomer has a constant Gibbs energy of formation D
involves a constant amount of energy EAA. We then have D
rG° = EAA for every reaction, with the D
gives D
create nor destroy monomers. We therefore have that
f Gi° = iD
ks
kd
¼ exp
EAA
kBT
;
which has the same value for every reaction, as desired.
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We now have a recipe to construct chemical reaction networks out of simple synthesis and
decomposition reactions. Despite the simplicity of the scheme, we will now show that the resulting
networks can exhibit surprisingly sophisticated autocatalytic mechanisms.
4 The Linear Case: Self-organization of First-Order Autocatalysis
In this section we present a simple nonequilibrium A-polymerization chemistry that can be solved
semi-analytically. This gives some insight into some sufficient conditions for exponential growth,
and the reasons why it emerges under these conditions.
In this chemistry we set r1,1
j = 1. That is, two monomers (A1) are not
allowed to directly join to form a dimer (A2), and, vice versa, a dimer cannot directly split into mono-
mers. All other reactions are permitted. There are multi-step reactions that can convert monomers
into dimers in this system, such as the autocatalytic cycle
= 0 and all other ri,
A1 þ A2 → A3;
A1 þ A3 → A4;
A4 → 2 A2;
(4)
→ 2 A2. (Here, as below, we use unidirectional arrows to
which has the net reaction 2 A1 + A2
represent the net direction in which these reversible reactions proceed.) This is called a first-order
autocatalytic cycle, because there is only one A2 on the left-hand side. First-order autocatalysis
leads to exponential growth, as we will see below. (See [7, 8] for background on the theory of
autocatalytic cycles, and [2] for a formal definition.)
Let us suppose that the initial conditions contain a very high concentration of A1 and trace
amounts of every other species. That is, a1 is large, and ai very small for i > 1. Under these assump-
tions, the rate of change of a1 will be small in comparison to its value, so that we can treat it as
constant. Using the approximation aiaj
= 0 for i, j > 0, Equation 2 becomes
da2
dt
dai
dt
X∞
(cid:2)
(cid:3)
¼ 2
kda2þj
− 2ksa1a2;
j¼1
Þkdai
(5)
¼ 2ksa1ai−1 − i−1
X∞
ð
(cid:2)
(cid:3)
þ 2
kdaiþj
j¼1
− 2ksa1ai
for i > 2. This may be written da/dt = Ra, where a is a column vector with elements a2, a3, . . ., and
R is the matrix
0
B
B
B
B
@
kd
−2K
2
2K −2K−2
0
0
⋮
2K
0
⋮
2
2
−2K−3
2K
⋮
2
2
2
…
…
…
−2K−4 …
⋮
…
1
C
C
C
;
C
A
(6)
with K = a1ks/kd.
It should be noted that although this approximation leads to a linear system of equations, it is
not a “near-equilibrium” approximation in the thermodynamic sense. The thermodynamic equilib-
rium of this type of system will in general contain a positive concentration of every species. We are
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Figure 1. The elements of the leading eigenvector of the matrix R, for K = 1.0 (top) and K = 100.0 (bottom). The
corresponding eigenvalues depend on kd, but are positive in both cases, meaning that the concentrations grow
exponentially while remaining proportional to the elements of the leading eigenvector. Both represent exponential
growth, but through different mechanisms.
linearizing the system around the state where most of the species are at zero concentrations; this
state is a dynamical fixed point, but it is very far from thermodynamic equilibrium.
Since this is a linear dynamical system, its asymptotic behavior is determined by the eigenvectors
and eigenvalues of R. In particular, since the only negative elements of R are on the diagonal, R has
only one eigenvector v with all positive entries (by application of the Perron–Frobenius theorem1 to
εR ≈ I + εR for sufficiently small ε), whose corresponding eigenvalue E is real.
the irreducible matrix e
Et, with C a real constant that depends on the initial
For sufficiently large t we have ai(t ) ≈ Cvi e
conditions. In the case where every ai = 0 (for i > 1), we have C = 0. For all other initial conditions,
C is positive.
The eigenvector and corresponding eigenvalue can readily be found by power iteration of e R Nð
Þ,
where R(N ) is an N × N matrix whose entries are the same as those of R (with the exception that
(N ) = −N, due to the absence of the reaction AN + A1
→ AN+1; for sufficiently large N this
RNN
makes very little difference).
Figure 1 shows two examples. In both, the leading eigenvalue of R is positive, meaning that the
species other than A1 will grow exponentially. In the case where K = 1, this growth is mostly due
to the minimal autocatalytic cycle shown in Equation 4, with a few side reactions producing longer
polymers.
1 The Perron–Frobenius theorem states that if a matrix (a ) has all non-negative entries and (b) is irreducible, meaning that the matrix can
be seen as the adjacency matrix of a strongly connected graph, then the following conditions hold (among others): (i) the eigenvalue with
the greatest magnitude is real, positive, and unique, and (ii) the eigenvector corresponding to this eigenvalue has all non-negative entries,
and is the only eigenvector with this property. It is a general property of the matrix exponential that the eigenvectors of eεR
will be the
same as those of εR, so we can conclude that R has only one eigenvector whose entries are all non-negative, which must then represent
the asymptotic distribution of the speciesʼ concentrations.
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However, the case with a larger K (meaning proportionally faster reaction rates in the synthesis
direction, or just a higher concentration of A1 initially) is dominated instead by larger molecules, and
its growth is therefore due to much larger autocatalytic cycles. There is no one cycle that dominates;
instead the microscopic picture is of a collection of polymers of many different lengths, which grow
at their ends due to monomer addition, but which also occasionally split somewhere in the middle,
giving rise to two new polymers.
It is a theorem in mass-action kinetics that there is a global Lyapunov function given by the
Gibbs energy, whose minimum is at the thermodynamic equilibrium point. Since we are linearizing
around a different fixed point, this guarantees that it will not be stable; at least one of its eigenvalues
must be non-negative. Because of this, there are no parameters that have to be tuned in order to
observe autocatalysis in this model. For any value of K there will always be a region around the point
a2
= a3
To derive these results we assumed that ai was small for i > 1, but since the concentrations of
these are increasing exponentially, this assumption will be violated eventually. This means that either
a1 will start to decrease at a substantial rate, changing the value of K, or terms of the form aiaj will
start to be important, changing the dynamics. In a real system, this may or may not happen before
the exponential growth becomes manifest enough to be clearly observable.
= . . . = 0 in which autocatalytic growth occurs.
It is worthwhile examining the assumptions we used to obtain this result. One important part of
⇌ A2 is removed from the reaction network. The reason for
the model is that the reaction 2A1
this is that this reaction would rapidly produce enough A2 that the assumption of small a2 would be
violated. We propose that in general any such direct route to product formation must be absent, or
at least occur only slowly, in order for autocatalysis to be observed.
The other important assumption was that the polymer species have (initially) very low concen-
trations, so that the aiaj terms vanish. Physically, this corresponds to a solution so dilute that two
polymer molecules are vanishingly unlikely to meet and react. (A1 monomers, however, are highly
concentrated, so the probability of reacting with a monomer is high.) Under these conditions,
exponential growth can occur through the molecules (a) growing into larger molecules by reacting
with monomers and (b) spontaneously splitting into two smaller molecules.
Such dynamics can be expected in much more complex thermodynamically reasonable chemis-
tries, as long as these conditions are obeyed; our reasoning is applicable to a much broader class of
models than the one presented here. The chemistry need not take the form of A-polymerization.
This reasoning will hold as long as there is some set of food species that cannot directly react to form
products, and so long as there are no energetically irreversible sinks in the network. No autocatalytic
cycles can arise unless they already exist in the network of potential reactions, but our example
shows that for even such a simple chemistry as A-polymerization there are many such potential
loops. We suspect these conditions will be satisfied by many chemistries, as long as they are suffi-
ciently densely connected and all the reactions have some degree of reversibility.
The mechanism as described so far results in exponential growth. Equations 5 is a linear dynamical
system, and therefore the only possibilities are exponential growth, exponential decay, and neutral
stability. (Oscillation around the initial point is impossible, because it would require concentrations
to become negative.) The more general assumptions outlined in the previous paragraph will also lead
to linear dynamical systems, resulting either in exponential growth or decay. In the next section we
explore the importance of nonlinearity in chemical networks, and we demonstrate that with some
changes to the reaction network, self-organization can produce superexponential (or hyperbolic)
growth in a simple A-polymerization chemistry.
5 Nonlinearity and Superexponential Growth
The constituents of a living cell are not capable of exponential growth at low concentrations; one
of the many functions of the cell membrane is to keep the concentrations high enough for growth to
occur. If the components of a living cell were to be placed in a dilute solution, it would be unlikely
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for an enzyme to collide with its substrate in order to bind to it, and therefore enzyme-catalyzed
reactions would occur at vanishingly low rates.
Exponential autocatalysis is interesting from an origins-of-life point of view [15], precisely be-
cause it doesnʼt require high concentrations, and therefore could exist before the evolution of
the cell membrane or other forms of compartmentalization. However, the linear analysis in the pre-
vious section makes it clear that the concentrations will always converge towards the same profile,
regardless of the initial conditions (as long as they obey the low-concentration assumption). This
seems to leave no room for evolution: There is no way one exponential replicator can outcompete
another.
Many models of prebiotic chemistry are nonlinear, because they involve reactions where two
non-food molecules must collide with each other. These include all models based on explicit catal-
ysis, such as the work of Kauffman [6] or the hypercycle model of Eigen and Schuster [4].
For these reasons it is worth exploring under which circumstances nonlinear, superexponential
growth can emerge spontaneously. In this section we demonstrate circumstances in which this can
happen in A-polymerization chemistries. The key is to constrain the reaction network so that first-
order, exponential autocatalytic cycles cannot occur. Under these circumstances, growth occurs if we
assume that reactions of the form aiaj (for i, j > 0) cannot occur. However, when integrating the
model numerically, we find autocatalytic cycles that rely on these reactions. Their kinetics are non-
linear (in fact superexponential), because they involve these quadratic terms. We show that if the
system is modeled as a flow reactor, this nonlinearity can give rise to behaviors such as bistability and
oscillations.
The exponential growth in the previous section occurs because a molecule can split into two
shorter molecules, each of which can then grow through monomer addition until it reaches the
→ 2An. Now we ask what
length of the original molecule, giving a net reaction An + nA1
happens if we prevent this simple first-order autocatalysis mechanism while retaining many of the
possible reactions from the original network. We show that autocatalysis still self-organizes, but it
does so through a more complex, second-order mechanism that results in superexponential growth.
In order to prevent this, we constrain the reaction network in the following way: We define a set
⇌ Ai+j is included in the reaction network unless either Ai 2
of banned species B. A reaction Ai + Aj
B, Aj 2 B, or Ai+j 2 B.
This technique of banning species is merely a convenient way to prevent the formation of first-
order autocatalytic cycles while retaining much of the original network; it is not supposed to directly
represent something one would expect to happen in polymerization chemistry. However, in more
complex chemistries a species may be stiochiometrically possible but have such a high free energy
that it is never formed, and so networks with broadly this type of structure might not be entirely
absent from real chemistry. We suspect that there are also many other reasons why a complex net-
work would fail to contain first-order loops.
To see how banning species can prevent first-order autocatalysis, let us imagine that B ¼
A2iþ1 : i 2 ℤ
g. That is, molecules whose lengths are powers of two are banned, apart from A1.
f
A molecule of An (with n not a power of two) may split into two smaller molecules, Am and
An−m, as long as neither m nor n − m is a power of two. However, at least one of these molecules
is no longer able to grow by monomer addition until it reaches length n. For let r ¼ 2½ log2n(cid:2) be the
largest power of two less than n. Then either m < r or n − m < r, since r ≥ n/2. A molecule of
size less than r cannot grow to size n by monomer addition alone, due to the absence of the reaction
→ 2An are impossible in this
Ar −1 + A1
chemistry.
→ Ar . Therefore net reactions of the form An + nA1
By excluding even more species we can obtain quite complex behavior. Here we present results
⇌ A2.
from a system that excludes the species B = {A3i : i 2 ℤ}, and also excludes the reaction 2A1
For computational reasons we also exclude species longer than 60 monomer units.
If, as in the previous section, we linearize this system around the point where all non-food
species have zero concentration, we find that the leading eigenvalue is zero. In such cases the
linear stability analysis is inconclusive, and the stability of the fixed point is determined by the
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Figure 2. The results of numerically integrating an A-polymerization chemistry over time. This chemistry excludes the
species B = {A3i : i 2 ℤ} ∪ {Ai : i > 60}, and in addition excludes the reaction 2A1 ⇌ A2. The parameters used are ks =
−5. (Many species
kd = 1. The initial concentration of A1 is 100, and the initial concentrations of all other species are 10
initially decay to much lower concentrations, only to recover later.) The curves appear roughly exponential on a
logarithmic scale, indicating superexponential growth occurring through higher-order autocatalysis. When plotted on
a linear scale, most speciesʼ concentrations appear negligible until the sudden transition to the equilibrium state at
around t = 330.
nonlinear terms. The theoretical apparatus for dealing with such cases is called center manifold
theory. However, for our purposes we can simply note that we know there is a global Lyapunov
function (the Gibbs energy) whose minimum is at a different point, the thermodynamic equilib-
rium. We should therefore expect the nonlinear terms to result in a second-order instability around
this fixed point, regardless of the details of how this comes about. Our numerical results in this section
show that this is indeed the case.
Figure 2 shows the results of numerically integrating this system, using Equation 2 with no
approximations, from an initial condition consisting of a high concentration of A1 and small
amounts of everything else. The result is that at around t = 450 there is a sudden transition to
the equilibrium state, in which all species are present, exponentially distributed according to their
lengths. These dynamics indicate that the species other than A1 grow (super )exponentially at low
rates until their concentrations become high enough for the growthʼs effects to be felt. This is sim-
ilar to the results from first-order autocatalytic systems presented in [15]. However, the mechanism
is rather more complicated.
At time t = 100, the growth in the system is largely due to the following set of reactions:
A8 → 2 A4;
A4 þ A1 → A5;
A4 → 2 A2;
A5 þ A2 → A7;
A7 þ A1 → A8:
(7)
These reactions may be stiochiometrically balanced to form the autocatalytic net reaction 3A8 +
→ 4A8. (That is true even though these are not the only reactions in the system, and they
8A1
are not necessarily happening at stoichiometrically balanced rates, because the concentrations are
changing over time. See Table 1 for the true reaction rates.) The nonlinearity comes from the
→ A7, which gives rise to a second-order nonlinear term because it requires
reaction A5 + A2
an A5 to collide with an A2 dimer in order to proceed; it will therefore increase in rate as the con-
centrations of these species increase.
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Table 1. The top ten reactions in the system shown in Figure 2, at time 100.
Reaction
A4 + A1 → A5
A7 + A1 → A8
A8 → 2A4
A4 → 2A2
A5 + A2 → A7
2A5 → A10
A10 + A1 → A11
A11 → A7 + A4
A13 + A1 → A14
A10 → A8 + A2
Rate
3.194 × 10
−5
1.888 × 10
−5
1.882 × 10
−5
1.366 × 10
−5
1.084 × 10
−5
8.050 × 10
−6
8.021 × 10
−6
7.942 × 10
−6
9.351 × 10
−8
8.598 × 10
−8
Notes. The rates shown are absolute net rates, in moles per unit time.
By t = 400 the system is dominated by the following overlapping but different set of autocatalytic
reactions, as can be seen in Table 2:
A11 → A7 þ A4;
A7 þ A1 → A8;
A8 → 2 A4;
A4 þ A1 → A5;
2 A5 → A10;
A10 þ A1 → A11:
(8)
→ 3A11. Where the
This is a second-order autocatalytic cycle with the net reaction 2A11 + 11A1
cycle in Equation 7 produces A7 by splitting A4 into A2 and then joining it with A5, this network
instead produces A7 by forming and then splitting A11. The transition from Equation 7 to 8 is prob-
→ A10 more
ably the result of the general increase in concentrations, making the reaction 2A5
→ A10 is the second-order reaction that gives rise to a
likely to proceed. In this network, 2A5
nonlinear term.
We reiterate that this nonlinear autocatalytic increase in concentrations is a logical consequence
of the reaction networkʼs structure. The values of the numerical parameters are more or less arbi-
trary; the only tuning required is to set the initial concentrations small enough for the effect to be
observed.
This nonlinearity can lead to bistability, which is important because it provides an analogue of
biological death, which in turn is necessary in order for one system to be able to outcompete
another. Suppose that additional reactions are added such that the polymer species decay expo-
nentially (and irreversibly) into inert products. Then at low concentrations the decay reactions will
be faster than the autocatalytic cycle and the polymer species will not be able to persist in the sys-
tem. However, at higher concentrations the autocatalytic species will grow at a faster rate than
their decay. Thus the polymer species can persist in the system only if they are initially present
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Table 2. The top ten reactions in the system shown in Figure 2, at time 400.
Reaction
A4 + A1 → A5
A7 + A1 → A8
A8 → 2A4
A10 + A1 → A11
2A5 → A10
A11 → A7 + A4
A5 + A2 → A7
A4 → 2A2
A13 + A1 → A14
A8 + A5 → A13
Rate
3.745 × 10
−3
1.549 × 10
−3
1.429 × 10
−3
1.202 × 10
−3
1.150 × 10
−3
1.088 × 10
−3
3.669 × 10
−4
2.466 × 10
−4
7.950 × 10
−5
3.669 × 10
−5
Notes. The overall difference in magnitudes compared to Table 1 is due to increases
in concentrations over time.
in concentrations above a certain threshold. This is in contrast to the case of exponential growth,
where the zero concentrations state is either a stable or an unstable fixed point, depending on
whether the growth rate of the autocatalytic cycle is greater or less than its decay rate. If the
zero-concentrations point is unstable, a single molecule can start the growth process and will even-
tually dominate the system.
The existence of a concentration threshold leads to a kind of precariousness that is an important
property of lifelike systems, since it is analogous to biological death [13, 3]. Precariousness means
Figure 3. Convergence to a limit cycle due to a higher-order autocatalytic cycle in a flow reactor. The reaction network
is the same as for Figure 2. The flow reactor is modeled with the flux terms fl = 0.01 (10000 − a1) and fi = −0.4ai
for i > 1. The parameters used are ks = 1.0 and kd = 1500. The autocatalysis is primarily due to the reactions shown in
Equation 7.
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there are some perturbations from which a system cannot recover; in this case some perturbations
push the concentrations below the threshold, and the concentrations must then eventually decay to
nothing.
This same kind of nonlinearity can also lead to a more easily visualized phenomenon, namely
temporal oscillations, as shown in Figure 3. To obtain this result, the reaction is modeled as if in
a flow reactor. There is a constant input of A1 and exponential decay of every other species, which
can be thought of as leaving the system in the reactorʼs outflow. The oscillations occur through a
similar mechanism to the famous lynx–hare cycle in ecology: The autocatalytic cycle consumes too
much of its food, and as a result its constituent species decay somewhat. This gives the food
concentration a chance to recover (since it is being continually fed into the system), and the auto-
catalytic species once again increase in concentration, completing the cycle.
This oscillatory behavior does not occur for every choice of parameter values. However, it
demonstrates nicely that the nonlinearity caused by higher-order autocatalysis can lead to phenom-
ena that are not possible in the linear, first-order case. We reiterate that all of the other results in this
article are very robust and require no parameter tuning at all.
6 Analysis and Conclusions
The second law of thermodynamics implies that physical systems must stochastically tend to “flow
downhill,” where “downhill” means in the direction of increasing entropy, or decreasing free energy.
We have implemented this feature in a very simple class of chemical reaction networks and found
that it can lead to self-organization of complex autocatalytic cycles.
This is in some way analogous to the self-organization of dissipative structures in physical sys-
tems, such as convection cells. These also involve the formation of cycles, and also increase the rate
at which energy is dissipated.
Our informal, intuitive understanding of these results is as follows: Under some (but by no
means all) circumstances, physical systems far from equilibrium not only “flow downhill,” but also
tend to seek out faster and faster ways to do this. The formation of convection cells is again an
example of this; the breaking of a dam is perhaps a more vivid one. The full A-polymerization
reaction network (with all reactions included) will flow to its equilibrium rather rapidly. However,
⇌ A2 reaction), then an indirect route must be found,
if we block the most direct route (the 2A1
by taking advantage of catalytic cycles. Blocking the zeroth-order route leads to the self-organization
of first-order autocatalytic cycles, because (in a very informal sense) these lead to faster rates of
dissipation.
Blocking these first-order solutions then leads to higher-order solutions. It is as if the system tries
to find the easiest route towards its equilibrium, and only once the simplest paths are blocked does it
find the more complex ones. We were surprised by the elegance and novelty of the networks that
emerged in the system, especially in comparison with second-order networks that we had tried to
design by hand.
From a more formal point of view, these results occur because the initial conditions are near
a fixed point that is not the thermodynamic equilibrium point, and therefore cannot be stable.
Creating such a far-from-equilibrium fixed point was simply a case of removing the direct reaction
from the network, so that monomers could not join together to form polymers in the absence of
other polymers. This suggests that many reaction networks should share this property of exhibiting a
non-equilibrium fixed point close to an experimentally realizable initial condition. We expect auto-
catalysis to be empirically observable in such situations.
By preventing first-order instability, we forced the initial fixed point to exhibit a second-order
instability. The nonlinearity of the resulting systems leads to phenomena such as precariousness
and oscillations. A nonlinear system with many equilibria is more interesting from an origins-of-life
point of view, because it is closer to a system in which multiple different autocatalytic systems can
compete with one another, leading to evolution.
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It is interesting to note that we found complex behavior in this system by removing reactions
rather than by adding them. If we consider the full A-polymerization chemistry, including both forward
and reverse reactions, all of these autocatalytic cycles—both first- and higher-order—are already
present as its subnetworks. Whether the more complex networks become manifested or not is a
matter of whether the right constraints exist to prevent simpler networks from being manifested instead.
Many authors in the origins of life have worried about the problem of side reactions that can pre-
vent autocatalysis from occurring (e.g., [11, 8, 14]). Our results suggest that under the right energetic
conditions, such issues might not arise. Our reaction networks contain many possible reactions be-
sides the ones that eventually form the autocatalytic cycles, yet these do not seem to disrupt the
autocatalysis to any meaningful extent. The reason is that any molecules produced by side reactions
eventually end up participating in autocatalytic cycles of their own. We therefore make the prediction
that when one complex autocatalytic cycle is observed experimentally, it will tend to be accompanied
by many others.
A closely related issue in the origins of life is the combinatorial explosion that can arise from
several small molecular components combining in different ways. Our model is incomplete in regard
to this issue—we have monomers of only one type, and they combine only linearly—but never-
theless our results suggest that the combinatorial explosion might not be as serious as it might at
first seem. One can imagine a class of molecules that can grow in multiple ways, leading to a com-
binatorial explosion, but that can also undergo reactions that decompose them into smaller mole-
cules. With such a chemistry, at low concentrations one should still expect exponential growth
through the mechanism that arises in the first of our models. We therefore suggest that the future
of origins-of-life models might be not in taming the combinatorial explosion [10] but in embracing
it. If every molecule forms part of an autocatalytic cycle, then the combinatorial explosion represents
an efficient way to explore the space of possible replicators, and it may be that it was through such a
process that the first high-fidelity replicators arose.
This research represents a step towards understanding what properties a chemical reaction
network must have in order for complex, lifelike dynamics to emerge. There is still a lot of work
to do to understand the effects of network topology and energetics on the emergence of complex
dynamics, but the results in this article represent a substantial step in the right direction.
Acknowledgments
We are grateful to Jade Foster for noticing some minor technical errors in the conference version of
this article. These errors have been corrected.
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