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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.

The authors have declared that no competing interests exist.

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

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

The language is based on the concept of boxes equipped with binders with π-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 compatibility of the binders, which is expressed through an affinity function applied to the type of the binder types

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

The chemoton

The metabolic subsystem produces the components necessary for its own self-reproduction and those of the other two subsystems. The membrane system provides compartmentalization 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 template system controls quantitatively the chemical processes of the whole supersystem

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

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

T_{m}…T_{m+k} represent the boundary subsystem, A_{1}…A_{5} represent the metabolic subsystem and pV(0)…pV(n−1) and pW(0)…pW(n−1) 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.

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

where |_{1}′ in the equation. The reactions are called for as events of the following form:

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 Gánti'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 _{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

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.

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.

where deletion rates are defined as:

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:

Template polymerization follows the method of

In the double-template model (_{1} and Z_{2}. Since Z_{1} and Z_{2} are represented as entities of constant concentration (assuming that the outside environment is stable and abundant in food), the proportion of V and W (and thus _{V} and _{W}) and backward reactions (_{V}′ and _{W}′), and the rates of polymerization (_{V6}, _{V7}, _{W6} and _{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.

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}, 20 _{m} and 1 Growth. Critical T_{m} is at 200. ∑A_{i} stands for the total amount of all metabolites, ∑_{i} for the total amount of all

The run was initialized with 200 A_{1}, 20 _{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.

_{V6} = _{V7} = _{W6} = 1, critical T_{m} = 1000. Top row: the polymerization rate of V (_{V7}) and W (_{W7}) are identical (1). Middle row: _{W7} = 10. Bottom row: _{W7} = 100. In the last two cases, _{V7} = 1. Volume and surface variables are omitted from the figure. Initial amounts: 100 A_{1}, 100 _{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 _{i}_{i}_{j}

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.

We also tested what happens if the external source of X is not constant, but has a low influx rate (

_{V6} = _{V7} = _{W6} = 1, critical T_{m} = 1000. Top row: the polymerization rate of V (_{V7}) and W (_{W7}) are identical. Middle row: k_{W7} = 10. Bottom row: _{W7} = 100. In the last two cases, _{V7}_{1}, 100 X, 100 _{m} and 1 Growth. Influx rate of X is 10; Z_{1} and Z_{2} are still constant.

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 _{1}→V or U+Z_{2}→W). 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.

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, ∑A_{i}_{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.

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 introduction 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.

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

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 congruence: 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.

We are grateful for the invaluable comments and discussions of Ferenc Jordán, Chrisantha Fernando, Balázs Könnyű and Gergely Boza. We are also deeply thankful for Lorenzo Dematté for helping with the BlenX language.