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Two Different Template Replicators Coexisting in the
Same Protocell: Stochastic Simulation of an Extended
Chemoton Model
Istva
´n Zachar
1
, Anna Fedor
1
*,Eo
¨rs Szathma
´ry
2,3,4
1HAS Theoretical Biology and Ecology Research Group, Department of Plant Taxonomy and Ecology, Eo
¨tvo
¨s University (ELTE), Budapest, Hungary, 2Department of Plant
Taxonomy and Ecology, Eo
¨tvo
¨s University (ELTE), Budapest, Hungary, 3Collegium Budapest (Institute for Advanced Study), Budapest, Hungary, 4Parmenides Foundation,
Pullach, Germany
Abstract
The simulation of complex biochemical systems, consisting of intertwined subsystems, is a challenging task in
computational biology. The complex biochemical organization of the cell is effectively modeled by the minimal cell
model called chemoton, proposed by Ga
´nti. Since the chemoton is a system consisting of a large but fixed number of
interacting molecular species, it can effectively be implemented in a process algebra-based language such as the BlenX
programming language. The stochastic model behaves comparably to previous continuous deterministic models of the
chemoton. Additionally to the well-known chemoton, we also implemented an extended version with two competing
template cycles. The new insight from our study is that the coupling of reactions in the chemoton ensures that these
templates coexist providing an alternative solution to Eigen’s paradox. Our technical innovation involves the introduction of
a two-state switch to control cell growth and division, thus providing an example for hybrid methods in BlenX. Further
developments to the BlenX language are suggested in the Appendix.
Citation: Zachar I, Fedor A, Szathma
´ry E (2011) Two Different Template Replicators Coexisting in the Same Protocell: Stochastic Simulation of an Extended
Chemoton Model. PLoS ONE 6(7): e21380. doi:10.1371/journal.pone.0021380
Editor: Andrey Rzhetsky, University of Chicago, United States of America
Received February 2, 2011; Accepted May 29, 2011; Published July 19, 2011
Copyright: ß2011 Zachar et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: Results have received funding from the European Community’s Seventh Framework Program (FP7/2007-2013) under grant agreement no. 225167
(project e-Flux). Support by the COST D27 action (prebiotic chemistry and early evolution) and COST CM0703 (systems chemistry) are also gratefully
acknowledged. The views expressed in this publication are the sole responsibility of the authors and do not necessarily reflect the views of the European
Commission. Additional support was provided by the National Office for Research and Technology (NAP 2005/KCKHA005) and by the National Scientific Research
Fund (OTKA 73047). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: fedoranna@gmail.com
Introduction
The simulation of complex biochemical systems, consisting of
intertwined subsystems, is a challenging task in computational biology.
The complex biochemical organization of the cell is effectively
modeled by the minimal cell–model called chemoton,putforwardby
Ga´nti [1,2,3,4]. In this paper we show an application of the process-
algebra based programming language BlenX [5,6,7] for implement-
ing the chemoton as a stochastic model of a complex chemical system.
BlenX was developed for modeling systems whose basic step of
computation is a monovalent event, or multivalent interaction
between subcomponents [6]. As such it is well suited for modeling
elementary or complex chemical reactions [6], Lotka-Volterra type
predator-prey dynamics [5] and community dynamics [8].
The language is based on the concept of boxes equipped with
binders with p-like processes inside. In the chemical reactions
example, boxes represent different molecules, whereas binders
express their interaction capabilities and processes handle the
manipulation of the binders and drive the internal behavior of the
boxes. In classical process calculi, boxes can interact provided that
they have identical channel names. BlenX, being a specific type of
process calculus itself, has its interactions guided by the compati-
bility of the binders, which is expressed through an affinity function
applied to the type of the binder types [7].
We used the BlenX language to implement the chemoton, and
the BetaWB simulator (v2.0.2) to run experiments. The stochastic
simulation engine implements an efficient variant of Gillespie
algorithms [5] that generates a trajectory of the stochastic
evolution of the system: it calculates which reaction will occur
next and when it will occur [9]. The key point of the method is to
use reaction probability per unit time instead of rate constants.
The algorithm used by the BetaWB simulator (called the Next
Action Method) uses both Gillespie’s Direct method and the First
reaction method [9].
The chemoton [1,2,3] is an autocatalytic chemical supersystem
that satisfies the criteria of life [1,4]. It consists of three autocatalytic
subsystems: a metabolic subsystem (self-reproducing chemical
cycle), an informational subsystem (template polymerization cycle)
and a boundary subsystem surrounding the former two (mem-
brane). The chemical reactions of the three subsystems are coupled
stoichiometrically, which coordinates the growth and division of the
cell thus helping the system to be autocatalytic as a whole.
The metabolic subsystem produces the components necessary
for its own self-reproduction and those of the other two
subsystems. The membrane system provides compartmentaliza-
tion and keeps the volume of the sphere between certain
boundaries whereby it ensures the necessary concentrations which
in turn are necessary for the appropriate rate of reactions. The
PLoS ONE | www.plosone.org 1 July 2011 | Volume 6 | Issue 7 | e21380
template system controls quantitatively the chemical processes of
the whole supersystem [3]. By the introduction of an information
carrier template molecule, at least limited heredity [10] can be
achieved: when the chemoton reaches a certain size, it splits into
daughter spheres, thus it is able to pass on changes in the template
molecules to offspring. In the basic model the template consists of
only one type of monomer, V, and the only real information the
system carries is the distribution of polymers of different lengths.
Although information is limited, it is still information.
The number of template variants can be further increased by
the addition of another monomer molecule, W. The different
monomers can form separate homopolymers an assumption that
helps us state the new findings as clearly as possible. Two
coexisting templates in a chemoton would be a step toward a
larger search space for evolution and toward a qualitative control
by the template subsystem over the whole chemoton instead of a
quantitative one.
However, internal competition of replicators (in this case
templates or polymers) poses a serious problem that arises from
the catch 22 of the origins of life: Eigen’s paradox. According to the
paradox no large genome can be maintained against errors without
enzymes, while no enzymes can exist without a large enough
genome [11]. One solution proposed the cooperative existence of
small template replicators that together may hold enough
information (the hypercycle [12]), although it was proven to be
insufficient without compartmentation. The stochastic corrector
model [13] provides a different solution by enclosing multiple
templates (unlinked genes) in compartments. It assumes that
replicative templates are competing within reproducing compart-
ments, whose selective values depend on the balance of internal
template composition. Stochasticity in replication and compartment
fission ensures that the fittest compartment types recur, allowing
therefore the stable coexistence of competing replicators. Thus
selfish (i.e. faster growing) mutants cannot destroy the system
causing the extreme dilution and extinction of slower templates.
We raise the possibility that internal competition may be solved
by other means within the chemoton; hence competing templates
can coexist within its boundaries. In the chemoton it is the
topology of coupling between metabolism and template replication
that can maintain coexistence. Our version of the chemoton makes
this link explicit by defining reversible reactions linking cell growth
with template growth.
In this paper we describe the behavior of two different models.
The first model is the standard chemoton model, based on a well-
studied set of continuous and deterministic standard nonlinear
kinetic differential equations [14]. We show with the help of the
BlenX language that a stochastic model behaves in the same way.
In the second model (Figure 1) the metabolic cycle produces two
different kinds of monomers which form two different kinds of
templates in two separate informational subsystems. We show that
these two seemingly competing templates (having exponential
growth trends a priori) can coexist in the system in spite of different
polymerization rates.
Methods
The chemoton is implemented as a series of events, each
representing an elementary reaction step (i.e. reversible reactions
are split to forward and backward elementary reactions).
Components are defined as boxes and are identified by their
binder types. Division of the chemoton is initiated and terminated
deterministically, but all other processes are stochastic.
The actual growth rate of each component is defined by their
kinetic constant k, times the amount of the reactants (with
appropriate exponents) producing it. The calculated growth rate
function is then plugged into the specific event representing the
reaction, as the rate of the event. The kinetic factors of a reversible
reaction:
A1zX
k1
k10
A2
are defined as follows:
let k1 : const = 2.0;
let k1r : const = 0.1;
let rate1 : function = k1 * |A1| * |X|;
let rate1r : function = k1r * |A2|;
where |n| indicates the cardinality (amount) of component nin the
system and k1r in the code equals to k
1
9in the equation. The
reactions are called for as events of the following form:
when(A1, X :: rate1) join(A2);
when(A2 :: rate1r) split(A1, X);
Since division is a global event that cannot be modeled from
molecule to molecule, but only concerning the cell as a whole, we
decided to implement it as a non-stochastic process. Division is
controlled by a global switch, which starts the growing or the
dividing phase of the cell. The standard chemoton reactions are
suspended during cell division. Although switching cell states is
deterministic and happens instantaneously, the division of
amounts is still a stochastic process as will be explained below.
In Ga´nti’s models cell division is triggered automatically when
the surface area of the cell doubles. Even if more realistic scenarios
exist (for example, in a stochastic model [15] the trigger is the
changing osmotic pressure) we chose to use the original
assumption for the sake of simplicity. Thus, in our model, cell
division always initiates when the surface of the membrane
(actually, the amount of T
m
molecules incorporated into the
membrane) reaches a certain, predefined size. At this point cell
division starts deterministically by switching the cell from the
phase of cell growth to the phase of cell division. (For a smoother
splitting mechanism, see [16], where the precise volume is
calculated depending on the shape of the cell during division.)
The two-state switch (acting as a global signal) is implemented by
boxes called Growth and Division:
when(Growth : |Tm| = 1000 : inf) split(Division, Nil);
when(Division : |Tm| = 500 : inf) split(Growth, Nil);
The two-state switch is a previously undocumented addition to
the BlenX armament of tools. It is a deterministic operator that
can be extended to handle more than two states as well. By
referring to the state of the switch, multiple events can be triggered
immediately. While the referred component (T
m
here) is properly
handled, no infinite loops should occur.
All the stochastic events are conditional on the presence of these
signals: the basic chemical reactions of the chemoton are only
active when the Growth signal is present; when the Division signal
is present all these reactions freeze until cell division is finished.
when(A1, X : |Growth|.0 : rate1) join(A2);
Conversely, division has been implemented as a set of
eliminating reactions which are only active when the Division
signal is present. Cell division is modeled by halving all molecular
amounts in the system. Only the halving of the amount of T
m
membrane molecules is deterministic and precise, all other
molecules are deleted with their specific deletion-rate until the
membrane reaches its post-division size, that is, until T
m
reaches
its lower bound.
when(Growth : |Tm| = 1000 : inf) split(Division, Nil);
when(Tm : |Division|.0 : delrateTm) delete(1);
Stochastic Template Competition in Protocells
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when(A1 : |Division|.0 : delrateA1) delete(1);
when(A2 : |Division|.0 : delrateA2) delete(1);
… // delete every other component
when(Division : |Tm| = 500 : inf) split(Growth, Nil);
where deletion rates are defined as:
let factor : const = 10.0;
let delrateA1 : function = factor * |A1|;
let delrateA2 : function = factor * |A2|;
…
The food molecule X either has a constant concentration, or is
constantly added to the system with a specific (fast) rate, which sets
the pace for the chemoton, for example:
let influx : const = 10.0;
…
when(X : |X| = 0 : influx) new(1);
Template polymerization follows the method of [14]. Initially a
double-stranded homopolymer of length n/2 is present (it consists
of two n/2-length polymer strands; nmonomers in sum). This state
of the polymer is called pV(0), indicating that zero extra monomer
was added to it so far above the basic n. During polymerization the
double stranded polymer is growing by successively adding extra V
molecules (e.g. pV(0)+VRpV(1)+R) until the total number of V
molecules in the polymer reaches 2n21. This state is denoted
pV(n21). Adding a further V to the polymer, it splits into two
pV(0) molecules, initiating thus further autocatalytic template
cycles. In each of the models discussed here n= 6. We entirely
ignored the polycondensation threshold value (usually present in
chemoton models, e.g. [3,14,17]), because it is not necessary in our
model for maintaining the growth and division cycles of the
chemoton. The mechanism of replication is deliberately kept as a
black box: we take a worst-case approach by assuming that
replication can result in exponential growth. We are aware of the
complication that non-enzymatic replication of nucleic acid
templates in general is an unsolved problem [18], but here we
address a different issue.
In the double-template model (Figure 1), the template
monomers V and W have a common precursor, U. Whether U
will be converted to V or W in reversible reactions depends
entirely on the availability of food molecules Z
1
and Z
2
. Since Z
1
and Z
2
are represented as entities of constant concentration
Figure 1. Chemoton with two templates. T
m
…T
m+k
represent the boundary subsystem, A
1
…A
5
represent the metabolic subsystem and
pV(0)…pV(n21) and pW(0)…pW(n21) represent two different template polymerization cycles (informational subsystems), T
1
and T
2
.Z
1
,Z
2
and X are
food molecules. See text for further details.
doi:10.1371/journal.pone.0021380.g001
Stochastic Template Competition in Protocells
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(assuming that the outside environment is stable and abundant in
food), the proportion of V and W (and thus pV and pW) will be
regulated by the kinetic factors of the forward (k
V
and k
W
) and
backward reactions (k
V
9and k
W
9), and the rates of polymerization
(k
V6
,k
V7
,k
W6
and k
W7
).
In the simple model, there is only one template polymerization
cycle which is directly connected to the metabolic cycle. There is
no precursor (U), food molecules Z1 or Z2; V is directly produced
by metabolism.
Note that the time scale of simulations is dimensionless; with the
proper setting of rates and initial amounts, the behavior scales
appropriately with time.
Results and Discussion
Basic model
Figure 2 shows the behavior of the basic chemoton in BlenX,
with similar kinetic constants as used by [14,17,19]. These
deterministic models agree that the chemoton maintains stable
cell cycles in various conditions. However, in Munteanu and Sole´’s
model [20], small changes in the concentration levels may lead to
drastic changes in the replication period. The cause of these
differences is still unclear. In the only stochastic model of the
chemoton [21] stable cell cycles were found. Our model also
confirmed these results.
The run was initialized with 200 A
1
,20pV(0), 10 T
m
molecules
and 1 Growth signal (and either 200 X and an influx rate, or a
constant amount of X). The critical amount of T
m
was 200:
division initiates when the number of T
m
molecules grows above
200. Right after division, the cell membrane contains 100 T
m
molecules. The initial concentrations need not be close to the
adapted concentrations of the chemoton, as the system self-
regulates: each component, independently of their initial amount,
reaches its typical value, which is maintained throughout the
oscillations. The chemoton can exist stably, with clockwork-like
oscillations. Its internal processes are synchronized to the division
process, which solely relies on the amount of membrane
molecules.
Double template model
Figure 3 shows now a second template subsystem can coexist
with the first one. We investigated the effect of different
polymerization rates on this model. In Figure 3 top row the two
templates have identical polymerization rates, and both V and W
are created in reactions with identical kinetic constants. As
expected, the two templates stably coexist.
In the second experiment, one of the polymers has a higher
polymerization rate. Usually, if an autocatalytic entity has a higher
growth rate than another one, its amount increases (due to the
autocatalytic nature of the template) to a point where the other
template is practically diluted to extinction. On the contrary, in
the chemoton two homopolymers can stably coexist, even if they
have different individual growth rates. Figure 3 middle and bottom
rows show that templates stably coexist when pW has a
polymerization rate 10 or even 100 times higher than that of
pV, respectively. In the chemoton all subsystems are stoichiomet-
rically coupled, meaning that the growth of each component is
synchronized with the overall growth of the chemoton.
We also tested what happens if the external source of X is not
constant, but has a low influx rate (Figure 4). This introduces both
Figure 2. Stochastic behavior of the chemoton. A: The food molecule X has an initial amount of 200 and is constantly added to the system with
a low rate (10). B: The influx rate of X is increased (200). C: X has a constant amount (10), representing a stable outside world. Runs were initialized
with 200 A
1
,20pV(0), 10 T
m
and 1 Growth. Critical T
m
is at 200. gA
i
stands for the total amount of all metabolites, gpV
i
for the total amount of all pV
polymer stages.
doi:10.1371/journal.pone.0021380.g002
Stochastic Template Competition in Protocells
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Figure 3. Dynamics of the chemoton with two different templates,
p
V and
p
W, when the concentration of food molecules is
constant. k
V6
=k
V7
=k
W6
= 1, critical T
m
= 1000. Top row: the polymerization rate of V (k
V7
) and W (k
W7
) are identical (1). Middle row: k
W7
= 10. Bottom
row: k
W7
= 100. In the last two cases, k
V7
= 1. Volume and surface variables are omitted from the figure. Initial amounts: 100 A
1
, 100 pV(0), 100 pW(0),
100 T
m
and 1 Growth. X has constant amount at 20, Z
1
and Z
2
at 10. Note that since division is set to a 100 times slower than in Figures 2 and 4,
removal of molecules is actually slower than growth. This has no effect on the outcome of the simulation, as the removal process is deterministic. gA
i
stands for the total amount of all metabolites, gpV
i
and gpW
j
for the total amount of all pV and pW polymer stages, respectively.
doi:10.1371/journal.pone.0021380.g003
Figure 4. Dynamics of the chemoton with two different templates,
p
V and
p
W, when the main food molecule (X) has a low influx
rate. k
V6
=k
V7
=k
W6
= 1, critical T
m
= 1000. Top row: the polymerization rate of V (k
V7
) and W (k
W7
) are identical. Middle row: k
W7
= 10. Bottom row:
k
W7
= 100. In the last two cases, k
V7
= 1. Volume and surface variables are omitted from the figure. Initial amounts: 100 A
1
, 100 X, 100 pV(0), 100 pW(0),
100 T
m
and 1 Growth. Influx rate of X is 10; Z
1
and Z
2
are still constant.
doi:10.1371/journal.pone.0021380.g004
Stochastic Template Competition in Protocells
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extra stochasticity, and a reduced pace for the oscillatory cycles
(note time scale), but still the experiments yield similar results.
Although the whole template subsystem is still stoichiometrically
coupled with metabolism and membrane growth the two
templates have internal dynamics. These are independent of the
dynamics of the chemoton as a whole: the total amount of pV(0)
and pW(0) polymers is the same as the amount of pV(0) in case of
the basic model. The internal template-competition is the
consequence of the OR type branching of the reactions of U
(U+Z
1
RVorU+Z
2
RW). The reversible nature of the reactions
producing V and W and the periodic fission of the chemoton
ensures that the faster template cannot wipe out the slower one.
To test the robustness of the chemoton, we have investigated
different metabolic subsystems as well. Figure 5 compares the
behavior of the model with m= 12 metabolites with m=5
metabolites: the difference is almost undetectable, indicating that
the model is capable of handling larger autocatalytic cycles as
well. Smaller metabolic subsystems can be crafted, though
somewhat forcibly: Ga´nti has defined the metabolic subsystem
of five members as the minimal cycle consisting of elementary
reaction steps. Fewer metabolites means that some of the
elementary reaction steps should be combined, yielding thus
non-elementary reactions. This, having no effect on the
simulation at all, blurs the clarity of the original chemoton
model, thus we mention it only briefly that results for m= 3 ar e
almost identical to those for m=5.
Conclusions
We have shown in this paper how to implement a stochastic
version of the chemoton with the help of the BlenX programming
language. BlenX was developed for implementing systems that
can be built up by the basic interactions of components; thus,
chemical reactions are quite straightforward to implement in this
language. Since the chemoton is a theoretically important and
well-studied system, this study could be a milestone for BlenX
applications.
A specific, somewhat contradictory solution was the introduc-
tion of a deterministic two-state switch in our BlenX code. It might
seem strange to introduce a deterministic process in a stochastic
model; however, it was very important for controlling the timing of
the different stages of the cell cycle, namely, the growth and the
division of the chemoton, being thus a prime example of hybrid
methods available in BlenX.
The new theoretical result of our model is that two competing
templates can coexist in the chemoton thanks to the topology of
the coupling of its chemical reactions. This finding suggests that
the constraint on coexistence of different templates, as assumed in
the formulation of Eigen’s paradox, may generally be more
relaxed than previously thought. Results are robust for both
polymer-size and the size of the metabolic subsystem.
Notes and suggestions for BlenX
The most important next step would be to allow the chemoton
to host a large set of possibly interacting templates, yielding
heteropolymers as well. This would open the scene for novel
combinations and also for mutations (i.e. hereditary variations) to
appear. By this way the evolution of templates could be introduced
into the model. However, evolution cannot be directly modeled in
BlenX. Evolution requires the random generation of sufficient
variability but at present, there is no method or function in BlenX
that could generate random variation (we are aware of the fact
that this is work in progress). At present, the only way to work
around is to manipulate the results of successive BlenX simulations
by external scripts [22]. A random number generator therefore
should be integrated with box/process/interface creation, to allow
the generation of new, programmable boxes on the fly, or the
mutation of existing boxes, processes, binder types, and binder
affinities during runtime.
This also requires the referencing of boxes or processes that
were not declared prior to running. An alternate naming system
could rely on parental relations rather than structural congru-
ence: each entity could be tracked, even if their internal structure
is unknown, by their parent entities. We conclude that the BlenX
language is a very good medium to simulate closed systems, with
predefined actors, such as predator-prey dynamics, or chemical
reactions. However, as evolutionary biologists, we would like to
see a more convenient way to model evolutionary systems as
well. The addition of the random number generator, its
integration with process-generation, and the possible referencing
of undeclared entities would open the door for evolutionary
modeling in BlenX, and possibly would give a boost for the
productivity of the language by making it a very useful tool in a
number of fields.
Provided that these improvements are present in a future release
of BlenX, real evolutionary simulations will become feasible with
the existing powerful capabilities of the language. Simulating more
advanced chemoton models in a stochastic way in BlenX would
yield important insights about the coexistence of replicators in
compartments, something which has been a holy grail for
researchers of prebiotic evolution for decades.
Figure 5. Comparison of different metabolic subsystems. The chemoton on the left consists of a 5-member metabolic cycle (A
1
to A
5
), while
the chemoton on the right harbors a 12-strong metabolism (A
1
to A
12
). The extra metabolites feed on X and the previous metabolite, and produce
the next metabolite in the cycle. The larger number of metabolic partners slightly decreases the total amount of metabolites, gA
i
. This is a
phenomenon that is supported directly by the numerical results of deterministic models: the larger the number of intermediates in the metabolic
cycle the less the total amount of metabolic molecules is in a splitting equilibrium. It is a consequence of the relative position where T
m
is produced
in the cycle: the earlier it is generated (i.e. the more metabolites are in the cycle after T
m
is generated), the less the total amount of metabolites will
be, as T
m
defines the critical value for splitting.
doi:10.1371/journal.pone.0021380.g005
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Acknowledgments
We are grateful for the invaluable comments and discussions of Ferenc
Jorda´n, Chrisantha Fernando, Bala´zs Ko¨nnyu˝ and Gergely Boza. We are
also deeply thankful for Lorenzo Dematte´ for helping with the BlenX
language.
Author Contributions
Conceived and designed the experiments: AF IZ ES. Performed the
experiments: AF IZ. Analyzed the data: AF IZ ES. Wrote the paper: AF IZ
ES.
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