# Design principles, growth laws, and competition of minimal autocatalysts

Yann Sakref  Olivier Rivoire  Gulliver, CNRS, ESPCI, Université PSL, Paris,
France.

###### Abstract

The apparent difficulty of designing simple autocatalysts that grow
exponentially in the absence of enzymes, external drives or ingenious internal
mechanisms severely constrains scenarios for the emergence of evolution by
natural selection in chemical and physical systems. Here, we systematically
analyze these difficulties in the context of one of the simplest and most
generic autocatalysts: a dimeric molecule that duplicates by templated
ligation. We show that despite its simplicity, such an autocatalyst can
achieve exponential growth autonomously. This only requires that the rate of
the spontaneous dimerization, the interactions between molecules, and the
concentrations of substrates and products are in appropriate ranges. We also
show, however, that it is possible to design as simple sub-exponential
autocatalysts that have an advantage over exponential autocatalysts when
competing for a common resource. We reach these conclusions by developing a
general theoretical framework based on kinetic barrier diagrams. Besides
challenging commonly accepted assumptions in the field of the origin of life,
our results provide a blueprint for the experimental realization of elementary
autocatalysts exhibiting a form of natural selection, whether on a molecular
or colloidal scale.

The path from simple chemical systems to complex living organisms is believed
to hinge on a pivotal point at which one molecule, or a set of molecules, gain
the capability to catalyze their own formation, hence constituting an
autocatalytic system orgel_molecular_1992 ; eigen_steps_1996 ;
luisi_emergence_2006 ; zachar_new_2010 ; hanopolskyi_autocatalysis_2021 . When
several such systems are formed from a common molecule, the faster ones hinder
the growth of the slower ones, which can lead to their exclusion. This
elementary form of natural selection is thought to set the stage for Darwinian
evolution orgel_molecular_1992 ; eigen_steps_1996 ; luisi_emergence_2006 .
Mathematically, exclusion occurs whenever replicators grow exponentially using
a common limiting resource, in which case only the fastest growing replicator
can possibly survive szathmary_sub-exponential_1989 ; scheuring_survival_2001
; kiedrowski_selection_2005 .

Molecular replication in extant living organisms relies on enzymatic catalysis
and involves a large network of coupled reactions. Non-enzymatic autocatalysts
have been designed in a variety of artificial systems and at a variety of
scales, from the molecular and colloidal scale up to the macroscopic scale
von_kiedrowski_self-replicating_1986 ; zielinski_autocatalytic_1987 ;
tjivikua_self-replicating_1990 ; wang_self-replication_1997 ; lee_self-
replicating_1996 ; issac_approaching_2002 ; kamioka_autocatalysis_2010 ;
colomb-delsuc_exponential_2015 ; kosikova_two_2019 ; zhuo_litters_2019 . At
the molecular scale, the simplest systems implement template replication,
where the formation of a new complex $AB$ from its constituents $A$ and $B$ is
catalyzed by a previously formed complex $AB$. However, such non-enzymatic
molecular autocatalysts are generally found to exhibit sub-exponential growth,
where the number $x$ of autocatalysts follows the phenomenological equation
$dx/dt=kx^{n}$ with $n<1$, associated with polynomial growth, $x(t)\sim
t^{1/(1-n)}$ dugas_minimal_1993 ; hanopolskyi_autocatalysis_2021 . A growth
order of $n\approx 1/2$ is typically observed, also known as parabolic growth
due to the relationship $x(t)\sim t^{2}$ von_kiedrowski_self-replicating_1986
; sievers_self-replication_1998 ; dugas_minimal_1993 ;
hanopolskyi_autocatalysis_2021 . Von Kiedrowski showed that this sub-
exponential growth originates from product inhibition, the propensity of
autocatalytic templates to inhibit their catalytic activity by binding to each
other dugas_minimal_1993 . Sub-exponential autocatalysts are not mutually
exclusive and are therefore often regarded as implementing only a primitive
and weak form of selection, if they are taken into account at all in the
emergence of natural selection dugas_minimal_1993 ; szathmary_replicators_1997
; scheuring_survival_2001 ; colomb-delsuc_exponential_2015 .

For this reason, much work has been devoted to the design of autocatalysts
that alleviate product inhibition. The first type of solutions involve
external drives applied in a cyclical pattern, such as heat braun_pcr_2004 ;
wang_self-replication_2011 ; zhuo_litters_2019 , mechanical stress
carnall_mechanosensitive_2010 , light zhang_engineering_2007 ;
zhang_accelerated_2014 , tidal cycling fernando_stochastic_2007 , or magnetic
fields dempster_self-replication_2015 . Approaches based on the intrinsic
properties of the autocatalyst rather than external factors have also been
proposed penrose_self-reproducing_1957 ; penrose_mechanics_1958 ;
penrose_self-reproducing_1959 ; virgo_evolvable_2012 ; abe_destabilizing_2004
; osuna_galvez_traceless_2019 ; zhang_engineering_2007 ; cabello-
garcia_handhold-mediated_2021 ; juritz_minimal_2022 . For instance, at the
molecular level, the affinity between autocatalysts can be diminished by
coupling the formation of a bound within autocatalysts to the breaking of a
bound between autocatalysts abe_destabilizing_2004 ;
osuna_galvez_traceless_2019 , or by entropic mechanisms like toeholds and
handholds strand displacements when using nucleic acids zhang_engineering_2007
; cabello-garcia_handhold-mediated_2021 ; juritz_minimal_2022 . These
approaches, although effective in specific settings, raise several questions.
First, it is often ambiguous whether the proposed mechanism mitigates product
inhibition, the binding of two preformed autocatalysts, or accelerate product
release, the unbinding of a newly formed autocatalyst from a preformed
catalyst, which is known to impact the growth rate but not the growth order of
autocatalysis virgo_evolvable_2012 ; cabello-garcia_handhold-mediated_2021 ;
zhang_engineering_2007 ; michaelis_amplification_2014 . Second, these
mechanisms are often idiosyncratic to the context in which they are designed,
which limits the scope of their applicability. Finally, and perhaps most
importantly, the designs are generally intricate and fine-tuned, which defeats
the purpose of studying autocatalysis as a means to understand the accretion
of complexity.

This poses a major problem for origin-of-life scenarios based on
autocatalysis. Consequently, most scenarios currently focus on autocatalytic
networks composed of multiple molecules rather than single-molecule
autocatalysts kim_cross-catalytic_2004 ; lincoln_self-sustained_2009 ;
zeravcic_self-replicating_2014 ; semenov_autocatalytic_2016 ;
cafferty_robustness_2019 ; blokhuis_universal_2020 ; ameta_darwinian_2021 .
This orientation reflects the belief that these networks are more likely to
emerge spontaneously kauffman_cellular_1971 ; eigen_principle_1977 ;
eigen_steps_1996 ; szostak_optimal_2011 ; liu_mathematical_2018 ;
hordijk_history_2019 ; hordijk_molecular_2019 . However, networks raise
similar challenges sievers_self-replication_1998 ; hordijk_influence_2018 , as
well as posing new ones, e.g., the likely appearance of parasites
eigen_selforganization_1971 ; matsumura_transient_2016 .

In either case, whether based on a single species or a network of species, the
design of autocatalysts has so far remained in the realm of empirical studies.
In particular, no theoretical argument rules out the possibility of simple
autocatalysts growing exponentially or establishes the minimum requirements
that they must satisfy. Here, we propose to fill this gap by showing through a
systematic approach that simple and generic autocatalysts are designable,
although with limitations that we clarify. By simple, we mean autocatalysts
composed of very few elements (two) with no internal structure or internal
degree of freedom. By generic, we mean an entropic mechanism of autocatalysis
by proximity that is present in any chemistry or colloidal system subject to
thermal noise. Our approach is to treat autocatalysis as a special case of
catalysis – namely when the product is a catalyst – and to apply a previously
developed methodology to define, construct and optimize minimal catalysts
munoz-basagoiti_computational_2023 ; sakref_kinetic_2023 . However, this is
only a starting point: as we show, this methodology needs to be extended to
account for the constraints arising from the identity between products and
catalysts, which introduces a fundamental distinction between catalysis and
autocatalysis. As a result, we demonstrate that it is possible to design
simple generic autocatalysts that grow exponentially, but that it is equally
possible to design simple sub-exponential autocatalysts that out-compete them
in conditions of resource limitation.

##  I Methods

### Model

We study the design of autocatalysts $AB$ composed of two units $A$ and $B$
which catalyze their own formation through a templating reaction summarized by
$AB+A+B\to 2AB$ (Fig. 1A). Guided by simplicity, we consider for $A$ and $B$
spherical particles of same diameter $\sigma$, immersed in a thermal bath at
temperature $T$ within a two-dimensional box of dimension $L\times L$. For
illustration and to indicate the experimentally feasibility of our design, we
take inspiration from DNA-coated colloids rogers_using_2016 and present
numerical results using a short-range, pairwise potential with a reverse
barrier (Materials and methods). As represented in Fig. 1B, this potential
features a cutoff distance of $r_{c}=1.1\ \sigma$, and a minimum at $r_{\rm
min}=1.03\ \sigma$ wang_lennard-jones_2020 ; cui_comprehensive_2022 . Thus,
only two parameters are left to specify the interaction between two particles
of types $X$ and $Y$: the energy barrier for dimer association,
$\epsilon_{XY}^{+}$, and the energy barrier for dimer dissociation,
$\epsilon_{XY}^{-}$.

With two particle types, $A$ and $B$, we generally need six parameters to
specify the interaction potentials. We reduce this number to two by making
additional simplifying assumptions. First, we consider that the dimerization
reaction, $A+B\to AB$, is irreversible ($\epsilon_{AB}^{-}=\infty$), and that
the interaction between $A$ and $B$ is therefore described by a single
parameter $\epsilon_{AB}^{+}$, the association barrier. Second, we consider
that the interaction potentials between two $A$ or two $B$ are the same, with
same depth ($\epsilon_{AA}^{-}=\epsilon_{BB}^{-}$) and no association barrier
($\epsilon_{AA}^{+}=\epsilon_{BB}^{+}=0$), leaving a single parameter
$\epsilon_{AA}^{-}$, the interaction strength, to describe their interaction.
Also to simplify the analysis, we assume that no molecule of size larger than
four can be formed. As summarized in Fig. 1C, the model has a total of three
dimensionless parameters: $L/\sigma$, $\epsilon_{AB}^{+}/k_{B}T$ and
$\epsilon_{AA}^{-}/k_{B}T$ where $k_{B}$ is the Boltzmann constant. Without
loss of generality, we set $\sigma=1$ to define the length scale, and
$k_{B}T=1$ to define the energy scale. To these three physical parameters, we
must add the current concentrations of molecular species. Again for
simplicity, we assume that $A$ and $B$ have the same concentration $[A]=[B]$.
The only remaining parameter is then $[AB]$, the concentration of free
products, or $[AB]_{\rm tot}$, the total concentration of products, including
those in complex with other species.

Figure 1: Model for the design of minimal autocatalysts A. Autocatalytic cycle
in which particles $A$ and $B$ can attach to dimer $AB$ catalyzing their
dimerization. The scheme represents $A$ binding $AB$ before $B$ but the
reverse order is also possible. Green arrows indicate diffusion processes,
dependent on the area $L^{2}$. Red arrows represent dissociations of two
identical particles, dependent on the interaction strength
$\epsilon_{AA}^{-}$. Blue arrows represent association between distinct
particles, dependent on the interaction barrier $\epsilon_{AB}^{+}$. The
spontaneous reaction involves both diffusion and association and is indicated
by a two-colored arrow. B. Potentials by which particles interact (Materials
and methods). Between identical particles, the potential depth is
$\epsilon_{AA}^{-}$, and association is diffusion-limited. Between distinct
particles, the potential depth is very large (infinite), and association
limited by a barrier $\epsilon_{AB}^{+}$.

### Questions

In the context of this model, the questions raised in the introduction can be
formulated as follow: What are the physical parameters $L$,
$\epsilon_{AB}^{+}$, $\epsilon_{AA}^{-}$ and the chemical conditions $[A]$ and
$[AB]$ for (i) optimal autocatalysis, that is, leading to a maximal
acceleration of the dimerization reaction $A+B\to AB$ by a pre-existing $AB$?
(ii) exponential growth, $d[AB]_{\rm tot}/dt=k[AB]_{\rm tot}$? (iii) exclusion
of an alternative autocatalyst $AD$ sharing with $AB$ a common constituent
$A$?

### Approach

As an intermediate step towards the design of an autocatalyst $AB$, we first
consider a dimeric catalyst $C=A^{\prime}B^{\prime}$, which is distinguishable
from $AB$, but has identical physical properties
($\epsilon_{A^{\prime}B^{\prime}}^{+}=\epsilon^{+}_{AB}$ and
$\epsilon^{-}_{A^{\prime}A^{\prime}}=\epsilon^{-}_{B^{\prime}B^{\prime}}=\epsilon^{-}_{AA}$).
Studying the catalysis $C+A+B\to C+AB$ enables us to apply and extend the
methods previously developed to design a minimal catalyst for the reverse
reaction, the dissociation of $AB$ into $A+B$ munoz-
basagoiti_computational_2023 , and provides a basis for subsequently exposing
the nuances between catalysis and autocatalysis.

More precisely, we derive constraints on the design of minimal autocatalysts
in four steps, starting from standard catalysis in the simplest setting and
progressively introducing elements of feedback inherent to autocatalysis: (1)
We determine the conditions under which a dimer $C=A^{\prime}B^{\prime}$ can
accelerate the dimerization reaction $A+B\to AB$. This is done by comparing
the time for the spontaneous formation of a dimer $AB$ in the presence and in
the absence of a $C$ sakref_kinetic_2023 . (2) Next, given a catalyst
$C=A^{\prime}B^{\prime}$, we determine the conditions for its optimal
efficiency. This is done by minimizing the cycling time $T_{\rm cycle}^{0}$,
defined as the mean time taken by one $C=A^{\prime}B^{\prime}$ to turn a
substrate $A+B$ into a product $AB$.

Following previous work munoz-basagoiti_computational_2023 , we solve (1) and
(2) in conditions that are most favorable for catalysis, namely in the absence
of any product $AB$ sakref_kinetic_2023 . (3) One unique feature of
autocatalysis, however, is that it necessarily takes place in the presence of
products, since the catalyst is itself a product. Products generally cause
product inhibition, whereby a product binds a catalyst and inhibits its
activity. We first analyze the consequence of product inhibition in standard
catalysis, when the catalyst $C=A^{\prime}B^{\prime}$ differs from the product
$AB$ and show that it increases the mean cycling time to $T_{\rm cycle}=T_{\rm
cycle}^{0}+T_{\rm inhib}$ with an additional time $T_{\rm inhib}$ that depends
on the concentration $[AB]$ of products. (4) Finally, we apply the previous
results to $C=AB$ and highlight how autocatalysis departs from catalysis. In
particular, while for standard catalysis the rate of product formation is,
when assuming the spontaneous reaction to be negligible, proportional to the
concentration of catalysts, i.e., of the form $d[AB]_{\rm tot}/dt=k[C]$ with
$k=1/T_{\rm cycle}$, this is no longer the case for autocatalysis because
$T_{\rm cycle}$ depends on $[AB]$ with $AB=C$.

Figure 2: Conditions for catalysis of the dimerization $A+B\to AB$. A. Mean
times for the dimerization $A+B\to AB$ in the absence of
$C=A^{\prime}B^{\prime}$ (in blue) and for the dimerization $C(A+B)\to C(AB)$
when $A,B$ are kept attached to $C$ (in red). A necessary condition for
catalysis is $T_{A+B\to AB}>T_{C(A+B)\to C(AB)}$ sakref_kinetic_2023 . Since
the first time scales with the reaction area $L^{2}$ while the second is
independent of it, catalysis requires a sufficiently large value of $L^{2}$.
The lines are from the Markov model presented in the text and the bars from
MD. B. The catalytic efficiency of $C$ shows a maximum at an intermediary
value of the interaction strength $\epsilon_{AA}^{-}$, consistent with
Sabatier principle. The value of this maximum increases with the interaction
barrier $\epsilon_{AB}^{+}$. C. The catalytic efficiency for optimal
$\epsilon_{AA}^{-}$ increases both with the reaction barrier
$\epsilon_{AB}^{+}$ and with the reaction area $L^{2}$.

##  II Design principles for minimal autocatalysts (auto)catalysts

### Conditions for catalysis

To determine the conditions under which a dimer $C=A^{\prime}B^{\prime}$ can
cause the acceleration of the reaction $A+B\to AB$, we first consider a closed
system with only one particle $A$ and one particle $B$ and determine the mean
time $T_{A+B\to AB}$ for a dimer $AB$ to form munoz-
basagoiti_computational_2023 . We compare this time to $T_{C+A+B\to C+AB}$,
the mean time for $AB$ to form when a prospective catalyst $C$ is added.
Catalysis occurs when this later time is shorter than the former, that is,
when the relative catalytic efficiency defined by the ratio $T_{A+B\to
AB}/T_{C+A+B\to C+AB}$ is superior to $1$. In previous work, we showed that a
molecule acts as an (auto)catalyst in presence of multiple molecules $A$ and
$B$ only if it acts as one in presence of a single $A$ and a single $B$
sakref_kinetic_2023 .

A first necessary condition is for the dimerization onto the catalyst to be
faster than the spontaneous reaction in the bulk, i.e., $T_{C(A+B)\to
C(AB)}<T_{A+B\to AB}$ sakref_kinetic_2023 . As expected from Arrhenius
equation, we verify with molecular dynamics (MD) simulations that both these
times scale exponentially with the association barrier $\epsilon_{AB}^{+}$
when it is sufficiently large (Fig. S1): $T_{A+B\to AB}\approx
L^{2}e^{\epsilon_{AB}^{+}}$ and $T_{C(A{+}B)\to CAB}\approx
e^{\epsilon_{AB}^{+}}$. Catalysis therefore requires a minimal area $L^{2}$ to
occur. For the design at hand, we find that an area of $(L/\sigma)^{2}\gtrsim
50$ is necessary (Fig. 2A).

Assuming such sufficiently large area $L^{2}$, we next study the impact of the
two physical parameters, $\epsilon_{AB}^{+}$ and $\epsilon_{AA}^{-}$. To
extend this study beyond the range of parameter values accessible by MD, we
approximate the dynamics by a Markov model with five distinct states,
corresponding to the various states of bonding between the autocatalyst, the
monomers $A$ and $B$, and the product $AB$ (Fig. 1A). Formally, the catalytic
cycle is described by

$C+A+B\xrightleftharpoons[\rho_{-1}]{\rho_{1}}CA+B\xrightleftharpoons[\rho_{-2}]{\rho_{2}}C(A{+}B)\xrightarrow[]{\rho_{3}}C(AB)\xrightarrow[]{\rho_{4}}C+AB,$
(1)

closed by adding $C+A+B\xrightleftharpoons[\rho_{-0}]{\rho_{0}}C+AB$ to
represent the spontaneous reaction without any interaction with the catalyst.
Here we assume that $A$ and $B$ are equivalent and therefore do not
differentiate between $CA+B$ and $CB+A$. We also assume that release occurs in
a single step, which is a good approximation when $\epsilon_{AA}^{-}$ is
sufficiently large (Fig S2). We take the dependence of the rate on the
parameters to be given by

$\displaystyle\rho_{1}\approx 2L^{-2},$ $\displaystyle\quad\rho_{2}\approx
L^{-2},\quad\rho_{3}\approx e^{-\epsilon_{AB}^{+}},\quad\rho_{4}\approx
e^{-2\epsilon_{AA}^{-}},$ (2) $\displaystyle\rho_{-1}\approx
e^{-\epsilon_{AA}^{-}},\quad\rho_{-2}\approx 2e^{-\epsilon_{AA}^{-}}.$

Pre-factors can be introduced to obtain a better fit to the MD simulations
(Supplementary Materials and Fig. S3), but they have no major impact on the
results (Fig. S8) and are omitted here to simplify the presentation.

The catalytic efficiency depends both on the interaction strength
$\epsilon_{AA}^{-}$ and on the association barrier $\epsilon_{AB}^{+}$. For a
given association barrier $\epsilon_{AB}^{+}$, we observe an optimal
interaction strength $\epsilon_{AA}^{-}$ (Fig. 2B). This observation follows
Sabatier’s principle, which applies broadly to catalytic systems with no
internal degrees of freedom medford2015sabatier ; rivoire_how_2023 , and
states that an optimal interaction between a catalyst and its substrate must
strike a balance between too weak an interaction that cannot hold the
substrates until they react, and too strong an interaction that cannot release
the product rapidly.

A second observation is that larger association barriers $\epsilon_{AB}^{+}$
enable greater relative catalytic efficiencies $T_{A+B\to AB}/T_{C+A+B\to
C+AB}$ (Fig. 2B) This is again a generic feature: the larger the barrier for
the spontaneous reaction, the more potential for catalysis. In fact no
catalysis can occur if the barrier is too small munoz-
basagoiti_computational_2023 . Finally, increasing the reaction area also
increases the relative efficiency of the catalyst (Fig. 2C). This is simply
the consequence of increasing the mean time of the spontaneous dimerization
reaction in solution without changing the dimerization reaction on the
catalyst.

In summary, catalysis of the reaction $A+B\to AB$ is favored by a large
reaction barrier $\epsilon_{AB}^{+}$, a large reaction volume $L^{2}$ and a
particular, finite value of the interaction strength $\epsilon_{AA}^{-}$ that
depends on $\epsilon_{AB}^{+}$ and $L^{2}$.

### Optimal cycling time in the absence of products

Having determined the conditions under which a molecule $C$ acts as a
catalyst, we now analyze how the catalytic turnover depends on the
concentration of substrates $[A]=[B]$. To this end, we can ignore the
spontaneous reaction. As a first step, we also assume that products are
systematically removed so that $[AB]=0$. The rates of the elementary processes
along the cycle are formally obtained by replacing $L^{-2}$ by $[A]$ in Eq.
(2), to account for the possible presence of multiple substrates. The Markov
chain for the complete cycle can be represented graphically as a kinetic
energy diagram penocchio_kinetic_2023 (Fig. 3A). In this diagram, each of the
five states $i$ ($i=1$ for $C$, $i=2$ for $CA$, $i=3$ for $C(A{+}B)$, $i=4$
for $C(AB)$ and $i=5$ for $C+AB$) is represented at an energy level $G_{i}$
and successive states are separated by transition states at energy level
$G_{i}^{\ddagger}$, such that the differences of energies between states and
transition states report the rates as

$\rho_{i}=e^{-(G_{i}^{\ddagger}-G_{i})},\quad\rho_{-i}=e^{-(G_{i}^{\ddagger}-G_{i+1})}.$
(3)

In this representation, the mean cycling time has a simple expression
kozuch_what_2011 ; park_describing_2022 ; penocchio_kinetic_2023 ,

$T_{\rm cycle}^{(0)}([A])=\sum_{1\leq i\leq j\leq
4}e^{G_{j}^{\ddagger}-G_{i}}$ (4)

where the sum is over each pair $i\leq j$ of transition state $j$ following a
ground state $i$ and where the subscript $(0)$ indicates that no product is
present. This sum is typically dominated by its largest term so that

$T_{\rm cycle}^{(0)}([A])\approx e^{\max_{1\leq i\leq j\leq
4}(G_{j}^{\ddagger}-G_{i})}.$ (5)

The exponent defines the limiting barrier, also known as the energy span
kozuch_what_2011 ; kozuch_how_2011 , which is represented in kinetic barrier
diagrams by the largest difference of energy between successive – but not
necessarily consecutive – levels. This limiting barrier formalizes the
intuitive but problematic notion of “limiting step”, which takes only into
account successive levels kozuch_how_2011 . As we show below, reducing the
estimation of the mean cycling time to the determination of the limiting
barrier simplifies the analysis and the interpretation of the results without
changing qualitatively the conclusions.

Limiting barriers can be of two types, direct barriers between successive
states and indirect barriers between non-successive states. Direct barriers
report the mean time to perform one elementary transition. The dependence of
the direct barriers $G_{i}^{\ddagger}-G_{i}=-\ln\rho_{i}$ on the parameters is
given by Eq. (2),

$\displaystyle G_{1}^{\ddagger}-G_{1}$ $\displaystyle\approx-\ln[A]-\ln 2,$
(6) $\displaystyle G_{2}^{\ddagger}-G_{2}$ $\displaystyle\approx-\ln[A],$
$\displaystyle G_{3}^{\ddagger}-G_{3}$
$\displaystyle\approx\epsilon_{AB}^{+},$ $\displaystyle
G_{4}^{\ddagger}-G_{4}$ $\displaystyle\approx 2\epsilon_{AA}^{-}.$

The first two barriers describe the diffusion of substrates to the catalyst,
the third barrier the dimerization reaction on the catalyst, and the last the
release of the product.

The total cycling time is, however, more than the addition of these elementary
transition times. Indeed, once a state has been reached, the next elementary
transition may be a backward transition and not a forward one, corresponding
to a recrossing event. This is the origin of the indirect barriers between
non-consecutive states, given by

$\displaystyle G_{2}^{\ddagger}-G_{1}=$
$\displaystyle\ln\frac{\rho_{-1}}{\rho_{1}\rho_{2}}\approx-\epsilon_{AA}^{-}-2\ln[A]-\ln
2,$ (7) $\displaystyle G_{3}^{\ddagger}-G_{1}=$
$\displaystyle\ln\frac{\rho_{-1}\rho_{-2}}{\rho_{1}\rho_{2}\rho_{3}}\approx-2\epsilon_{AA}^{-}+\epsilon_{AB}^{+}-2\ln[A],$
$\displaystyle G_{3}^{\ddagger}-G_{2}=$
$\displaystyle\ln\frac{\rho_{-2}}{\rho_{2}\rho_{3}}\approx-\epsilon_{AA}^{-}+\epsilon_{AB}^{+}-\ln[A]+\ln
2.$

These indirect barriers are all the smaller than the backward direct barriers
– the direct barriers for the catalysis of the reverse reaction $AB\to A+B$ –
are higher. Hence, a short cycling time requires not only low forward direct
barriers but also high backward direct barriers. As apparent in Eqs. (6) and
Eqs. (7), the different barriers are not independent but controlled by the
same physical and chemical parameters. These relationships capture the
essential trade-offs involved in the design of catalysis.

Figure 3: Kinetic energy diagram and limiting barriers – A. Kinetic energy
diagram associated with the Markov chain described by Eq. (1) and Eq. (8).
Local minima represent states while local maxima represent transition states,
at levels corresponding to the rates between successive states, as given by
Eq. (3). The mean cycling time is approximated by the largest difference
between successive levels, as indicated in Eq. (5). In this illustration, it
is given by $G^{\ddagger}_{3}-G_{1}$ (light green) but other values of the
parameters can lead to other limiting barriers. As in Fig. 1, different colors
refer to different processes: light blue for dimerization, darker blue for
association of a substrate followed by dimerization, green for diffusion of
both substrates followed by dimerization, and orange for the dissociation of
the product. Backward direct barriers are indicated with dashed lines. In the
presence of products, an additional state $i=0$ can be reached, representing a
non-productive complex $\overline{C(AB)}$, here placed on the left of the
diagram. B. Limiting barrier as a function of the substrate concentration
$[A]$ and the interaction strength $\epsilon_{AA}^{-}$, for
$\epsilon_{AB}^{+}=10$, and no product, $[AB]=0$. The red line represents the
optimal interaction strength with three different regimes, (a), (b), (c),
depending on which two barriers are in trade-off (see Supplementary Material).

For instance, at low substrate concentration, the optimal interaction energy
is $\hat{\epsilon}_{AA}^{-}=(\epsilon_{AB}^{+}-2\ln[A])/4$. Consistent with
Sabatier principle, this optimum strikes a balance between the indirect
barrier for substrate binding and dimerization $G_{3}^{\ddagger}-G_{1}$ (in
green in Fig. 3B), which is diminished by increasing $\epsilon_{AA}^{-}$, and
the direct barrier for product release $G_{4}^{\ddagger}-G_{4}$ (in orange in
Fig. 3B), which is conversely diminished by increasing $\epsilon_{AA}^{-}$.
The same reasoning applies at higher substrate concentrations, where the
direct barrier for product release is in trade-off with other indirect
barriers related to substrate binding (segments (a) and (b) in Fig. 3B, see
Supplementary Material).

The analysis of limiting barriers in kinetic barrier diagrams thus reveals how
different trade-offs control the design of optimal catalysts, depending on
chemical and physical parameters (Fig. 3B).

### Optimal cycling time in the presence of products

We now extend the analysis to the presence of free products, $[AB]\neq 0$. The
presence of product generally increases the mean cycling time, because a
catalyst can bind to a product instead of a substrate, thus forming a non-
productive complex that we denote $\overline{C(AB)}$. This non-productive
complex $\overline{C(AB)}$ is physically indistinguishable from the unreleased
complex $C(AB)$ that constitutes the last step along a catalytic cycle (Fig.
1A) but the recognition that they are two different kinetic states is key to
our analysis. Since $\overline{C(AB)}$ is a complex with a previously free
$AB$, while $C(AB)$ is a complex with a newly made $AB$, they are indeed
associated with two distinct constraints on catalysis, namely product
inhibition and product release. Formally, Eq. (1) already accounts for product
release, and additionally accounting for product inhibition is done by
extending it to include

$\overline{C(AB)}\xrightleftharpoons[\rho_{-I}]{\rho_{I}}C+AB$ (8)

where $\rho_{I}\approx\rho_{-4}\approx[AB]$ and $\rho_{-I}\approx
e^{-2\epsilon_{AA}^{-}}$.

The total mean cycling time $T_{\rm cycle}([A],[AB])$ is then increased by the
mean time $T_{\rm inhib}([A],[AB])$ spent in the inhibited state
$\overline{C(AB)}$,

$T_{\rm cycle}([A],[AB])=T_{\rm cycle}^{(0)}([A])+T_{\rm inhib}([A],[AB]).$
(9)

The slowdown due to product inhibition is a particular form of competitive
inhibition where the product itself acts as the inhibitor bezerra_enzyme_2016
.

$T_{\rm inhib}([A],[AB])$ can be expressed by extending the kinetic barrier
diagram to include a state $i=0$ associated with $\overline{C(AB)}$, leading
to

$T_{\rm inhib}([A],[AB])=\sum_{i=1}^{3}e^{G_{i}^{\ddagger}-G_{0}},$ (10)

where the new kinetic barriers to consider are obtained from the previous ones
as

$G_{i}^{\ddagger}-G_{0}=G_{i}^{\ddagger}-G_{1}+\ln\frac{\rho_{I}}{\rho_{-I}}$
(11)

for $i=1,2,3$, leading to

$\displaystyle G_{1}^{\ddagger}-G_{0}$ $\displaystyle\approx
2\epsilon_{AA}^{-}-\ln[A]+\ln[AB]-\ln 2,$ (12) $\displaystyle
G_{2}^{\ddagger}-G_{0}$
$\displaystyle\approx\epsilon_{AA}^{-}-2\ln[A]+\ln[AB]-\ln 2,$ $\displaystyle
G_{3}^{\ddagger}-G_{0}$
$\displaystyle\approx\epsilon_{AB}^{+}-2\ln[A]+\ln[AB].$

As shown in Fig. 4A, those additional barriers can dominate the others,
leading the mean cycling time to be limited by product inhibition, $T_{\rm
cycle}\approx T_{\rm inhib}$. In particular, this happens for large relative
concentration of products, $[AB]\gg[A]$, such that the catalyst is more likely
to bind a product than a substrate, and for large interaction strength with
respect to the concentration of product, $\epsilon_{AA}^{-}\gg-\ln[AB]/2$,
such that the time spent in the inhibited complex $\overline{C(AB)}$ is long
(SM). Since the barriers associated with product inhibition increase with
$\epsilon_{AA}^{-}$, one consequence of the accumulation of products is
generally a decreased optimal interaction strength, as illustrated in Fig. 4A.

##  III Growth laws for minimal autocatalysts

### Production rate

Assuming a buffered concentration of free substrates $A$ and $B$, and a
negligible spontaneous reaction, the rate of product formation is obtained
from the mean cycling time as ninio_alternative_1987

$\frac{d[AB]_{\rm tot}}{dt}=\frac{1}{T_{\rm cycle}([A],[AB])}[C]_{\rm tot},$
(13)

where $[AB]_{\rm tot}$ is the total concentration of products, including those
which, after being formed, bind to a catalyst or a substrate, and where
$[C]_{\rm tot}$ is the total concentration of catalysts, either free or bound.
With standard catalysis, $[C]_{\rm tot}$ remains constant and the rate of
product formation is simply proportional to it. With autocatalysis, however,
$C=AB$, and the total concentration of catalysts increases as more products
are formed. Eq. (13) becomes

$\frac{d[AB]_{\rm tot}}{dt}=\frac{1}{T_{\rm cycle}([A],[AB])}[AB]_{\rm tot},$
(14)

which is generally a non-linear function of $[AB]_{\rm tot}$ since $[AB]$ is
itself a function of $[AB]_{\rm tot}$. Special conditions are therefore
required for exponential growth to occur, where $d[AB]_{\rm tot}/dt=k[AB]_{\rm
tot}$ with a rate $k$ independent of $[AB]_{\rm tot}$.

Figure 4: Limiting barriers in the presence of products. A. Limiting barriers
for a given concentration of free product $[AB]$. Compared to Fig. 3B, the
limiting barrier can be associated with product inhibition (regimes in darker
colors), in which case $T_{\rm cycle}\approx T_{\rm inhib}$. As a consequence,
the optimal interaction strength $\epsilon_{AA}^{-}$ is changed (red line). B.
Limiting barriers when fixing the total concentration of autocatalyst
$[AB]_{\rm tot}$ instead of the concentration of free autocatalyst $[AB]$. The
results are similar at low $\epsilon_{AA}^{-}$, when $[AB]_{\rm
tot}\simeq[AB]$, but different at large $\epsilon_{AA}^{-}$, when $[AB]_{\rm
tot}\simeq[(AB)(AB)]$. In these two cases, $[\overline{(AB)(AB)}]\ll[AB]_{\rm
tot}$, implying two opposite limits with no product inhibition and, therefore,
exponential growth. C. Reaction order $n$, as computed from simulations of the
ordinary differential equations describing the Markov model (Materials and
methods). In comparison to B, we see that $n<1$ even in regions where the
limiting barrier is not associated with product inhibition. This is because
product inhibition is always present, even when it does not control the
limiting barrier. A value $n>0.9$ is nevertheless observed for a large range
of parameter values (dark blue).

### Conditions for exponential growth

The decomposition of the cycling time in Eq. (9) makes explicit the conditions
for exponential growth to occur: since $T_{\rm inhib}([A],[AB])$ depends on
$[AB]$ but $T_{\rm cycle}^{(0)}([A])$ does not, we must have $T_{\rm
cycle}^{(0)}([A])\gg T_{\rm inhib}([A],[AB])$, i.e., product inhibition must
be negligible.

Fig. 4A shows that this occurs when release rates significantly exceed
diffusion rates, $\epsilon_{AA}^{-}\ll-\ln[A]$, which a systematic analysis of
limiting barriers confirms (Supplementary Material). Fig. 4A is drawn for a
fixed concentration of free product $[AB]$, but it is often more informative
to fix the total concentration of products, $[AB]_{\rm tot}$, which better
reflects the progression of the dynamics and the consumption of resources – a
determining factor when considering competitions as below.

When considering a fixed $[AB]_{\rm tot}$, Fig. 4B also shows that product
inhibition is negligible when $\epsilon_{AA}^{-}\ll-\ln[A]$. This coincides
with the results of Fig. 4A because in this case most products are free, i.e.,
$[AB]_{\rm tot}\approx[AB]$, which implies that inhibiting complexes
$\overline{(AB)(AB)}$ are negligible (Fig. S6). However, a significant
difference appears in the opposite limit $\epsilon_{AA}^{-}\gg-\ln[A]$ where,
in contrast to Fig. 4A, Fig. 4B shows an extended regime where product
inhibition is negligible (in orange). In this regime, most products are in the
form of unreleased complexes $(AB)(AB)$ and therefore also not forming
inhibiting complexes $\overline{(AB)(AB)}$ (Fig. S6). The distinction made in
Eq. (8) and Fig. 3A between the physically identical but kinetically distinct
states of unreleased and inhibiting complexes is critical to understand this
regime.

Reducing the analysis to the identification of limiting barriers is an
approximation that provides necessary but not sufficient conditions for
strictly exponential autocatalysis: a barrier associated with product
inhibition may indeed contribute significantly to the cycling time even if it
is not the limiting barrier. To go beyond this approximation, we approximate
the dynamics with the phenomenological equation $d[AB]_{\rm tot}/dt=k[AB]_{\rm
tot}^{n}$ (Materials and Methods) and analyze the conditions under which
$n\approx 1$. We verify that these conditions are more demanding than those
for which the limiting barrier is not associated with product inhibition, but
nevertheless verify that autocatalytic growth is nearly exponential growth for
a large number of parameters, even taking into account the constraint that the
growth rate must exceed the spontaneous reaction rate (Fig. 4C).

The conditions for exponential growth, either weak or large interaction
strengths, are in direct contrast to the conditions for minimal cycling time
which, following Sabatier principle, requires an intermediate interaction
strength (Fig. 2B). As illustrated in Fig. 4C, this translates into a generic
trade-off between the reaction constant $k$ and the reaction order $n$. The
strength of this trade-off depends, however, on the values of the interaction
barrier $\epsilon_{AB}^{+}$: increasing $\epsilon_{AB}^{+}$ mitigates this
trade-off, drawing the optimal values of $k$ and $n$ closer together (Fig.
S7). This occurs for large values of both $\epsilon_{AB}^{+}$ and
$\epsilon_{AA}^{-}$, when $\epsilon_{AB}^{+}>\epsilon_{AA}^{-}>-\ln[AB]/2$.
Indeed, as $\epsilon_{AB}^{+}$ increases, the optimal $\epsilon_{AA}^{-}$ also
increases according to Sabatier’s principle, until a point where it saturates
and where no free autocatalyst remains, thereby preventing product inhibition.

##  IV Competition rules for minimal autocatalysts

One consequence of product inhibition is that the cycling time alone does not
determine the outcome of competitions between autocatalysts. To demonstrate
this, we consider in Fig. 5 a simple setting with three autocatalysts in a
chemostat, $AB$, $AD$, and $AE$, all competing for a common resource $A$.
Substrates $A$, $B$, $D$, and $E$ are supplied at a uniform constant rate
$\tau^{-1}$, and all molecules are diluted at the same rate $\tau^{-1}$, so
that $\tau$ represents the typical residence time in the chemostat.

Previous theoretical investigations have emphasized a fundamental difference
between exponential ($n=1$) and sub-exponential ($n<1$) autocatalysts in such
conditions: while exponential autocatalysts invariably compete to exclude one
another, sub-exponential autocatalysts typically coexist szathmary_sub-
exponential_1989 ; scheuring_survival_2001 ; lifson_coexistence_2001 ;
kiedrowski_selection_2005 . In recent work, we considered the competition of
autocatalysts of different order $n$ and noted that, somewhat
counterintuitively, a sub-exponential autocatalyst ($n<1$) can exclude an
exponential one ($n=1$) if its reaction constant $k$ is sufficiently large
sakref_exclusion_2024 . This occurs, notably, at high dilution rates, when the
mean residence time of the molecules in the chemostat is short relative to the
mean cycling time, or, equivalently, when resources are scarce. In such
conditions, autocatalysts are kept at a low concentration, mitigating product
inhibition and making reaction constants $k$ the determining factor. This is
verified in our model when competing $AB$ with $AD$, an autocatalyst of higher
$n$ but lower $k$: $AD$ dominates $AB$ only for sufficiently large values of
$\tau$ (Fig. 5A). Thus, not only does an optimal cycling rate not guarantee
dominance, but no intrinsic property of the autocatalyst guarantees it
independently of the extrinsic conditions in which the competition takes
place. Finally, this figure also illustrates how multiple autocatalysts
competing for the same resource may either coexist or exclude each other,
despite no strict exponential growth ($n$ is never strictly $1$).

Figure 5: Competition for a common limiting resource. Steady-state
concentrations of non-competing (dotted lines) and competing (plain lines)
autocatalysts $AB$, $AD$, and $AE$ in a chemostat, as a function of the
residence time $\tau$. The steady-state concentrations are normalized by the
concentration at which the substrates are supplied,
$[A]_{0}=[B]_{0}=[C]_{0}=[D]_{0}=e^{-10}$. While low residence times favor
$AB$, higher residence times favor autocatalysts $AD$. The figure also
illustrates how an autocatalyst of a lower efficiency, here $AE$, can be
excluded.

##  V Discussion

Our analysis of minimal autocatalysis reveals that, contrary to what previous
attempts might have suggested, exponential autocatalysts can be designed
without recourse to complex internal mechanisms, complicated geometries, or
external drives. Past limitations are well-known to originate from product
inhibition, the propensity of autocatalysts to bind to each other after they
have been produced. Our analysis concurs in recognizing product inhibition as
a fundamental constraint, but indicates that this constraint can be
circumvented by an appropriate choice of parameters. Our approach also
clarifies how the limitations due to product inhibition differ from those due
to product release, which impacts the reaction rate but not necessarily the
reaction order.

The focus on exponential growth stems from the exclusion principle that it
implies, which is often considered as a core principle of natural selection
dugas_minimal_1993 ; scheuring_survival_2001 : two exponential autocatalysts
cannot coexist if they depend on the same resource. Our results underline that
exponentiality is not an intrinsic property of an autocatalyst, but crucially
depends on extrinsic conditions, and that exclusion can occur in the absence
of exponential growth. However, it can also be argued that the absence of
strict exclusion is in itself conducive to the emergence of diversity and of
evolution by natural selection szathmary_sub-exponential_1989 ;
zachar_new_2010 .

The key feature of our model is its definition based solely on physical
principles: all possible molecules and reactions are derived from interaction
potentials between elementary “atoms”. This reveals how different rate
constants are in trade-off because they depend on the same physical
parameters. Our analysis of simple competitions between autocatalysts can thus
go beyond previous studies where product inhibition is phenomenologically
described by a reaction constant $k$ and a reaction order $n$ szathmary_sub-
exponential_1989 ; lifson_coexistence_2001 ; scheuring_survival_2001 . In
particular, our model demonstrates how $n$ and $k$ can be in trade-off:
maximizing $n$ to achieve exponential growth ($n=1$) typically comes at the
expense of a low $k$. In competitions between autocatalysts, whether a large
$k$ or a large $n$ is advantageous depends on the chemical environment. If
resources are abundant, autocatalysts with higher reaction order tend to
prevail, but, if resources are scarce, autocatalysts with higher reaction
constants have an advantage, irrespectively of their reaction order. An
exclusive focus on the reaction order $n$ may therefore be misleading.

We defined a generic and simple model with a view to its implementation in
various molecular or colloidal systems. First, we chose the catalytic
mechanism to be of the most basic form: the (auto)catalyst catalyzes a
dimerization reaction simply by increasing the frequency of interaction
between substrates when they are attached to it. This form of catalysis by
proximity is universal and applies irrespective of whether the dimerization
barrier is entropic or enthalpic. The parameters in our model also have their
direct counterpart in almost all chemical contexts. For example, in the realm
of nucleic acids, inter-dimer interactions correspond to base pairing via
hydrogen bonds, while stronger intra-dimer interactions with an association
barrier correspond to nearly irreversible endothermic phosphodiester covalent
bonds monnard_taming_2016 ; dickson_determination_2000 . In the realm of
colloids whose interactions are mediated by the hybridization of complementary
DNA strands or by magnetic forces, association barriers can correspond to
electrostatic repulsion, to an entropic barrier due to steric effects, or to
linkage-mediated interactions dempster_self-replication_2015 ; angioletti-
uberti_mobile_2014 ; rogers_mean-field_2020 ; wang_colloids_2012 ;
feng_specificity_2013 ; zhang_multivalent_2018 ; kress_colloidal_2020 ;
mcmullen_self-assembly_2022 . In this context, interaction strengths are
typically of the order of a few $k_{B}T$ and unbinding occur within $<1$ min
rogers_kinetics_2013 ; cui_comprehensive_2022 . In this case, exponential
growth would require interaction strengths of the order of $10$ $k_{B}T$,
depending on the relative concentration of substrate over product (Fig. S8).
Exponential autocatalysts would then replicate within hours, in the range of
experimentally accessible timescales.

For the sake of simplicity, we assumed that no molecule larger than four in
size can form. This is straightforwardly the case with molecular systems that
are intrinsically anisotropic von_kiedrowski_self-replicating_1986 ;
zhuo_litters_2019 ; kress_colloidal_2020 but, may be more difficult to impose
on isotropic colloids angioletti-uberti_mobile_2014 . However, a simple
extension of the model translates this assumption into a constraint on the
valence of atoms that is easier to implement. Our analysis indeed applies
without change to the cross-catalysis of two dimers $AB$ and
$A^{\prime}B^{\prime}$, where each type of atom is constrained to interact
with at most two atoms of two different types, $A$ with $B$ and $A^{\prime}$,
$B$ with $A$ and $B^{\prime}$, $A^{\prime}$ with $A$ and $B^{\prime}$,
$B^{\prime}$ with $A^{\prime}$ and $B^{\prime}$. DNA or RNA replication works
by such cross-catalysis between complementary strands sievers_self-
replication_1998 ; zhang_engineering_2007 ; zhuo_litters_2019 . With spherical
colloids, cross-catalysis can for instance be implemented by limiting
interactions to patches kress_colloidal_2020 (Fig. S4). However, we
constrained the size of the molecules only to simplify the analysis, and the
possibility of forming larger molecules is obviously of interest on its own.

The trade-offs that constrain our model fundamentally stem from its deliberate
simplicity. In particular, the tension between chemical acceleration, on the
one hand, and product release and inhibition, on the other, which underlies
Sabatier principle and plays a key role in our analysis, can be overcome by a
variety of mechanisms perez-ramirez_strategies_2019 ; rivoire_how_2023 . In
all practical cases, however, these mechanisms involve large and complex
molecules. Our analysis shows that they are not prerequisites for exponential
growth, or selection by exclusion. This resolves an apparent paradox in
origin-of-life scenarios that seek to explain complexity as a consequence of
Darwinian evolution, but require complex mechanisms for such evolution to take
place.

## Acknowledgments

We acknowledge funding from ANR-22-CE06-0037.

## MATERIALS AND METHODS

### Molecular dynamics simulations

Brownian molecular dynamics (MD) simulations were carried out in HoomD 3.5.0
anderson_hoomd-blue_2020 , using a time step $\Delta t=10^{-5}$, periodic
boundary conditions, and a damping constant $\gamma=10$, corresponding to a
translational diffusion coefficient $k_{B}T/\gamma=0.1\text{
length}^{2}/\text{time}$, comparable to values measured in experiments with
colloids zhang_multivalent_2018 . The potential between two particles $X$ and
$Y$ is taken to be

$U_{XY}(r)=\begin{dcases}\epsilon_{XY}^{-}u\left(r\right)+\epsilon_{XY}^{+}&\text{if
}r\leq r_{\mathrm{c},}\\\
-\epsilon_{XY}^{-}u\left(r-r_{\mathrm{c}}+r_{\min}\right)&\text{if
}r_{\mathrm{c}}\leq r\leq 2r_{\mathrm{c}}-r_{\min}\end{dcases}$

where $\epsilon_{XY}^{-}$ and $\epsilon_{XY}^{+}$ represent the activation
barriers for dissociation and association, respectively. The potential $u(r)$
is a generalization of the Wang-Frenkel potential wang_lennard-jones_2020 ,
with a cutoff value of $r_{c}=1.1$,

$u(r)=\begin{dcases}\alpha\left(r_{\mathrm{c}}\right)\left[\left(\frac{\sigma}{r}\right)^{2}-1\right]\left[\left(\frac{r_{\mathrm{c}}}{r}\right)^{2}-1\right]^{2}&\text{for
}r\leq r_{\mathrm{c}}\\\ 0&\text{for }r>r_{\mathrm{c}}.\end{dcases}$

### Markov model

We approximate catalytic cycles by Markov chains. With two atoms $A$ and $B$,
the Markov chain involves a total of 15 transitions. First, the spontaneous
reaction $A+B\to AB$, with rate $[A][B]e^{-\epsilon_{AB}^{+}}$. Second, $7$
association reactions, $AB+A\to AAB$, $AB+B\to ABB$, $AAB+B\to AABB$,
$ABB+A\to AABB$, $AB+AB\to\overline{ABAB}$, $A+A\to AA$ and $B+B\to BB$, with
rates proportional to the reactant concentrations. Third, the dimerization
reaction on the autocatalyst, $AABB\to ABAB$, with rate
$[AABB]e^{-\epsilon_{AB}^{+}}$. Finally, $8$ dissociation reactions, $AA\to
A+A$, $BB\to B+B$, $AAB\to A+AB$, $ABB\to B+AB$, $AABB\to B+AAB$, $AABB\to
A+ABB$, with rates $e^{-\epsilon_{AA}^{-}}$, and $ABAB\to AB+AB$,
$\overline{ABAB}\to AB+AB$ with rates $e^{-2\epsilon_{AA}^{-}}$. When
considering two competing autocatalysts $AB$ and $AC$ sharing a common
monomer, we ignore for simplicity the complexes that they may form, of the
type $(AB)(AC)$, which are less stable than homotetramers $(AB)(AB)$.

We determine the steady state of the Markov chain by integrating numerically
the ordinary differential equations that describe its dynamical evolution. We
consider the system to be either closed (Fig. 4C) or in a chemostat (Fig. 5).
In this later case, differential equations include the description of the
introduction of substrates, $\emptyset\to A$ and $\emptyset\to B$, with rate
$[A]_{0}/\tau$, and the dilution of all species, $X\to\emptyset$, with rate
$1/\tau$.

### Reaction order and reaction constant

We estimate a reaction order $n$ and a reaction constant $k$ such that
$d[AB]_{\rm tot}/dt=k[AB]_{\rm tot}^{n}$ approximatively holds by integrating
numerically the dynamical equations of the Markov chain with constant values
of $[A]=[B]$, starting with $[AB]=0$ and ending when reaching the targeted
value of $[AB]_{\rm tot}$. The values of $k$ and $n$ are then obtained by
linear regression of $\ln(d[AB]_{\rm tot}/dt)$ against $\ln([AB]_{\rm tot})$.

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