Chemistry Department, University "Aldo Moro", Bari, 70125, Italy

Abstract

Background

Over the last two decades, lipid compartments (liposomes, lipid-coated droplets) have been extensively used as in vitro "minimal" cell models. In particular, simple and complex biomolecular reactions have been carried out inside these self-assembled micro- and nano-sized compartments, leading to the synthesis of RNA and functional proteins inside liposomes. Despite this experimental progress, a detailed physical understanding of the underlying dynamics is missing. In particular, the combination of solute compartmentalization, reactivity and stochastic effects has not yet been clarified. A combination of experimental and computational approaches can reveal interesting mechanisms governing the behavior of micro compartmentalized systems, in particular by highlighting the intrinsic stochastic diversity within a population of "synthetic cells".

Methods

In this context, we have developed a computational platform called ENVIRONMENT suitable for studying the stochastic time evolution of reacting lipid compartments. This software - which implements a Gillespie Algorithm - is an improvement over a previous program that simulated the stochastic time evolution of homogeneous, fixed-volume, chemically reacting systems, extending it to more general conditions in which a collection of similar such systems interact and change over the course of time. In particular, our approach is focused on elucidating the role of randomness in the time behavior of chemically reacting lipid compartments, such as micelles, vesicles or micro emulsions, in regimes where random fluctuations due to the stochastic nature of reacting events can lead an open system towards unexpected time evolutions.

Results

This paper analyses the so-called Ribocell (RNA-based cell) model. It consists in a hypothetical minimal cell based on a self-replicating minimum RNA genome coupled with a self-reproducing lipid vesicle compartment. This model assumes the existence of two ribozymes, one able to catalyze the conversion of molecular precursors into lipids and the second able to replicate RNA strands. The aim of this contribution is to explore the feasibility of this hypothetical minimal cell. By deterministic kinetic analysis, the best external conditions to observe synchronization between genome self-replication and vesicle membrane reproduction are determined, while its robustness to random fluctuations is investigated using stochastic simulations, and then discussed.

Background

In recent years, many researchers have been actively working in the field of the

The topic of artificial cells is strongly related to the minimal cell notion that defines the simplest cell to be considered alive and then to be experimentally implemented. A minimal living cell, or protocell, can be defined as the minimum supramolecular bounded structure based on the lowest number of molecular species and metabolic processes that is capable of self-maintaining, self-reproducing and evolving

Some years ago, Szostak and colleagues proposed a minimal cell prototype called the _{L}) able to catalyze the conversion of molecular precursors (P) into membrane lipids (L) and the other one (R_{P}) able to duplicate RNA strands. Therefore, in an environment rich in both lipid precursors (P) and activated nucleotides (NTPs), the Ribocell can self-reproduce if both processes, i.e. genome self-replication and membrane reproduction (growth and division), are somehow synchronized.

In previous papers _{L }^{3}s^{-1}M^{-1}≤_{L}≤1.7·10^{5}s^{-1}M^{-1 }then synchronization between vesicle reproduction and genome replication can spontaneously emerge under the model assumptions and kinetic parameters reported in Table _{L}

The Ribocell internal metabolism

**The Ribocell internal metabolism**. (1) Reversible RNA strand association, (2) catalyzed template transcription (S = R_{P}, _{c}R_{P}, R_{L}, and _{c}R_{L},) (3) lipid synthesis.

Kinetic Parameters for the

**Kinetic**

**Parameters**

**Values**

**Process Description**

**Ref**.

_{SS}[s^{-1}M^{-1}]

8.8·10^{6}

Formation of dimers R_{c}R_{P }and R_{c}R_{L}

_{S}[s^{-1}]

2.2·10^{-6}

Dissociation of dimers R_{c}R_{P }and R_{c}R_{L}

_{R@S}[s^{-1}M^{-1}]

5.32·10^{5}

Formation of R@S

_{R@SS}[s^{-1}]

9.9·10^{-3}

Dissociation of Complexes R@S_{c}S

_{NTP}[s^{-1}M^{-1}]

0.113

Nucleotide Polymerization in Oleic Vesicle

_{L }[s^{-1}M^{-1}]

1.7·10^{3}

Lipid Precursor Conversion*

_{in }[dm^{2}s^{-1}]

7.6·10^{19}

Oleic acid association to the membrane

_{out }[dm^{2}s^{-1}]

7.6·10^{-2}

Oleic acid release from the membrane

_{P }[cm·s^{-1}]

4.2 10^{-9}

Membrane Permeability to Lipid Precursor

_{NTP }[cm·s^{-1}]

1.9 10^{-11}

Membrane Permeability to Nucleotides

_{W }= _{S}

0.0

Membrane Permeability to W and genetic staff

_{aq}[cm·s^{-1}]

1.0·10^{-3}

Oleic Acid Membrane Permeability to Water

*_{L }is 10^{5 }times larger than the value of the splicing reaction, catalyzed by the hammerhead ribozyme.

In this paper, we first apply this model to 100-base-long ribozymes, in an attempt to find the best experimental conditions to reduce so as Ribocell life time. By using the deterministic approach, the robustness of the stationary growth and division regime will be investigated in terms of external substrate concentrations, vesicle size and initial ribozyme amount in order to define optimal external conditions for Ribocell self-reproduction.

Therefore, the influence of ribozyme length will also be explored in the optimal external conditions by ranging strand size from 20 to 200 bases in length and keeping all the other kinetic parameters constant. 20 bases is in fact the minimum length required to observe a folded RNA structures, i.e. a structure that can reasonably exhibit catalytic action. On the other hand, entities of about 200 nucleotides have been suggested as plausible ancient proto-ribosomes

Finally, stochastic simulations will be performed in order to test the robustness of the ribocell base on 100-base length ribozymes in optimal external conditions, with the aim of elucidating the role of intrinsic and extrinsic stochasticity on the time behavior of a protocell population.

Methods

Self-reproducing vesicles are compartmentalized chemically-reacting systems where self-assembly processes are coupled with chemical reactions that produce amphiphilic molecules. To study the time behavior of these systems, we use both a deterministic and a stochastic approach in order to get insights into the average behavior of the protocell population and, at the same time, to elucidate the role of random fluctuations. Given a certain minimal cell model, i.e. a reaction mechanism with all the required parameters (kinetic constants, permeability coefficients, initial concentrations), the deterministic analysis can be done by numerically solving the ordinary differential equation set (ODES) or by analytically integrating an approximated reduced set of differential equations. Examples of the latter approach can be found in our previous works where self-reproducing micelles

A chemically reacting vesicle can be described as a homogeneous reacting aqueous domain enclosed by a lipid bilayer. Molecules can be exchanged with the external environment thanks to transport processes through the lipid membrane. A flux of water can also take place through the membrane in order to balance the osmotic pressure, i.e. the difference between the internal and the external overall concentrations. Chemical reactions can occur in the vesicle water core, according to the assumed internal metabolism, and amphiphilic molecules can be absorbed from and released towards both the external and internal aqueous solutions. Hence, the vesicle time state is defined by the following array:

where _{i}^{C }are the molecular numbers of species X_{i }(_{L}^{μ }is the number of amphiphiles X_{L }_{C }is the water internal volume. In the stochastic approach, all _{i}^{C }and _{L}^{μ }are discrete integer numbers and there exist as many arrays as vesicles in the systems, while in the deterministic analysis there is a single array with real values that represents the average time state of the entire reacting vesicle population.

Table

Deterministic rates against propensity density probability for reacting and transport events.

**Event**

**Deterministic Rate**

**( Ms) ^{-1}**

**Propensity Density Probability**

**
s
^{-1}
**

Internal Chemical reactions ^{(a)}

Solute X_{n }membrane transport ^{(b)}

Membrane Lipid Release

Membrane Lipid Uptake

Water Flux ^{(d)}

^{a)}**a **and **b **stoichiometric matrixes, _{A }Avogadro's number, _{ρ }kinetic constant, _{ρ}molecularity

^{(b) }The relationship between the macroscopic permeability _{n }and the molecular diffusion coefficient _{n }is: _{n }= _{n}λ_{μ}_{A}, λ_{μ }being the membrane thickness.

^{(c) }The absolute value guarantees that the propensity density probability is positive and the molecules move in the opposite direction from the concentration gradient.

^{(d)}_{aq }is the water molar volume, while _{T}^{E }and _{T}^{C }are the total osmotic concentration in the external and internal aqueous solutions, respectively.

Self-reproducing vesicles

The vesicle surface is estimated by the formula: S_{μ }_{L}_{L}^{μ}_{L}_{L }

that is, the ratio between the actual membrane surface _{μ }and the spherical area that would perfectly wrap the actual core volume V_{C}

_{T}^{C}_{T}^{E}^{1/3}). So

The Ribocell model

Figure _{P }according to the steps in bracket (2). This process is described as a catalytic template-directed addition of mononucleotides with high fidelity and processivity. It starts with R_{P }binding any of the monomeric strands S (S = R_{P}, _{c}R_{P}, R_{L }and _{c}R_{L}) to form the complex R@S. This complex will then initiate the polymerization of the conjugate strand _{c}S, by coupling and iteratively binding the complementary bases and releasing the by-product W. When the strand _{c}S has been completely formed, the polymerase ribozyme releases the new dimer. Finally, the ribozyme R_{L }catalyzes the conversion of the precursor P into the lipid L (3).

Table _{L }and R_{P }are created with a random sequence of bases, and they are assumed to have similar kinetic behaviors for the sake of simplicity. The kinetic constants of formation _{SS }and dissociation _{S }of both dimers were set equal to the values experimentally measured for a sequence of 10 nucleotides _{c}S (S = R_{P}, _{c}R_{P}, R_{L }and _{c}R_{L}) were set equal to those measured for the human enzyme β-polymerase _{L }^{5 }times larger than that of the splicing reaction, catalyzed by the hammerhead ribozyme _{NTP }derived from experimental data simulations (De Frenza private communication) of the DNA template directed synthesis in fatty acid mixed vesicles _{L}^{2 }and the osmotic tolerance ε = 0.21 are defined according to data reported in literature for oleic acid vesicles _{W }= 0.0 cm/s, based on the assumption that W is a charged species, and the permeability to the precursor: _{P }= 0.42·10^{-8 }cm/s, corresponding to the oleic acid membrane's permeability to Arabitol

A common simplifying assumption to both approaches is to consider the external concentrations of nucleotides (NTPs), lipid precursor (P) and inert compound (B) to be constant throughout the time-courseof the process, thanks to an incoming flux of material in the reactor vessel:

As has been pointed out before, given the set of kinetic parameters reported in Table

In order to achieve this in our model, a spontaneous synchronization between membrane and the aqueous volume core of the self-replicating vesicle must take place. By introducing _{V }and surface _{S}:

it is easy to show(unpublished observations) for growing protocells that synchronization can take place only if

Nevertheless, it is worthwhile to keep in mind that _{S }is essentially proportional to the rate of lipid synthesis, since amphiphile uptake by the membrane is very fast when the concentration of lipids is above the equilibrium value. Instead, having assumed the external value _{T}^{E }to be constant and a vesicle being in a osmotic balanced condition: _{T}^{E}≈ _{T}^{C }= _{C}_{T}^{C},_{V }is driven by the overall internal _{T}^{C }population rise. Therefore, since _{T}^{C }increases essentially owing to the waste production that takes place with ribozyme self-replication and lipid synthesis, this has the effect of coupling the membrane reproduction with the genome replication and with the volume growth:

When the Ribocell reaches a stationary regime, at each division the genetic materials can be randomly distributed between the daughters. If the amount of genetic material is very low, then this can result in a separation of R_{P }from the other RNA strands. In fact, the Ribocell must contain a minimum genetic kit of three RNA filaments in order to be capable of self-replicating its entire genome: one R_{P }that catalyzes the RNA base pair transcription, one (R_{L }or _{c}R_{L}) and one (R_{P }or _{c}R_{P}) that work as templates for the transcription. Moreover, since R_{L }is necessary to catalyze lipid precursor conversion, the optimal minimum 3-ribozyme kit must be made up of 2R_{P }and one R_{L}. This minimum kit should be at least doubled before cell division, in order to have a chance that both daughters continue to be active. Therefore, if a random distribution of RNA filaments takes place after vesicle division, ribozyme segregation between the two daughters might occur. Different scenarios can be envisaged as sketched in Figure _{P }or many filaments of _{c}R_{P }and/or _{c}R_{L }(_{L }strands are _{P }or both R_{P }and _{c}R_{P }filaments are able to self-replicate this reduced genome (_{P }filament and R_{L}/_{c}R_{L }strands. As a consequence of this, _{L}/_{c}R_{L }genetic stuff, and at the same time to synthesize lipids. Therefore, they can grow and divide, producing in turn at least one reduced ribocell and/or self-replicating, inert and empty vesicle. With the help of stochastic simulations, we will try to explore all the possible scenarios.

Different-reacting protocells and vesicles obtained by RNA segregation due to Ribocell division

**Different-reacting protocells and vesicles obtained by RNA segregation due to Ribocell division**. Nucleotides (NTPs) and waste (W) have been omitted for the sake of clarity, along with the reversible association of RNA.

Results and discussion

Deterministic analysis

In the present paper, we firstly explore the dependence of the stationary regime on the external concentrations of substrates for ribozymes 100 nucleotides long setting _{L }to 1.7·10^{3}s^{-1}M^{-1}, i.e. the minimum value in the previously observed synchronization range. The aim of this preliminary deterministic study is to find the optimal initial conditions in order to achieve a stationary regime with the shortest life time. All the outcomes are reported in additional file

Deterministic Outcomes of the Ribocell time behavior: stationary values for different initial conditions

Click here for file

As first, the dependence of Ribocell state on overall external concentration _{T}^{E }is analyzed at the stationary regime, reached after 20 generations. Since [_{ex}] = [_{ex}] = 5.0·10^{-4}M, the overall external concentration can be approximated to _{T}^{E}≈[_{ex}]. The upper plots in Figure _{ex}] increases, then the Ribocell radius _{20 }decreases, while the life cycle Δ_{20 }rises. Thus, vesicles become smaller and more dormant as the overall external concentration rises. This can be ascribed to the mechanism of synchronization itself and is in agreement with what we reported in a recent work (unpublished paper) where an inverse dependence of the vesicle steady size on overall external concentration was explicitly derived from the general stationary condition _{ex}

Dependence of the Ribocell stationary regime on the external concentration of the inert compound [I_{ex}]

**Dependence of the Ribocell stationary regime on the external concentration of the inert compound [I _{ex}]**. Vesicle radius (left upper plot), division time (right upper plot), overall internal concentrations of RNA strands (left lower plot) and genome composition percentage (right lower plot) were determined after 20 generations.

Figure _{L }and _{c}R_{L }strands are present, both around 33%, while the percentages for R_{P}, and _{c}R_{P }are lower: 24% and 10%, respectively. Due to the high _{ss }value, the ribozymes are mainly present inthe form of dimers and this accounts for the equal fractions observed for the lipase ribozymes, while the percentage of R_{P }is greater than that of _{c}R_{P}, since some R_{P }strands are involved as catalyzers in template duplication. This also explains why the overall percentage of polymerase ribozymes (~34%) is lower than that of lipase ribozymes (~66%) in fact, not all polymerase ribozymes are available as templates for duplication. Moreover, this asymmetry is amplified as long as the total concentration of the genetic material increases, see data in additional file

Setting [_{ex}] = 0.3 M, we study the dependence of the stationary division regime on the external concentration of the substrates: lipid precursor [_{ex}] and nucleotides. The same concentration value [_{ex}] is set for the four different nucleotides since they have been assumed to have the same kinetic behavior. The upper plot of Figure _{ex}] and [_{ex}] on the stationary vesicle radius _{20}. Higher [_{ex}] concentrations speed up genome self-replication with respect to lipid synthesis, accelerating waste production and leading to larger core volumes. Conversely, increasing [_{ex}] reduces _{20 }since membrane self-reproduction becomes faster. For the same reason, the total concentration of genetic material is increased due to the high concentrations of nucleotides and the low concentrations of the lipid precursor, while, both substrate concentrations decrease cell life time when they are increased, since all the metabolic processes are accelerated. If [_{ex}] ≥ 0.05 M, the Ribocell undergoes an osmotic burst since volume growth is too fast compared to lipid production for any value of [_{ex}] in the studied range (see additional file

Dependence of the Ribocell stationary regime on the external concentration of nucleotides [N_{ex}] and lipid precursor [P_{ex}]

**Dependence of the Ribocell stationary regime on the external concentration of nucleotides [N _{ex}] and lipid precursor [P_{ex}]**. Vesicle radius (upper plot), division time (left lower plot) and logarithm of the overall internal concentrations of RNA strands (right lower plot) were determined after 20 generations.

In the upper plots of Figure _{0 }= 100 for both R_{c}R_{L }and R_{c}R_{P}. These plots show that after a few generations the steady division regime is reached, as confirmed by _{0}. In all three cases, the same division time is reached after 10 generation:s, i.e. 68.2 days, that remains constant for the following generations. The higher N_{0}, the faster the cell division in the first generations.

Deterministic time evolution of the Ribocell

**Deterministic time evolution of the Ribocell**. Aqueous core volume (left upper plot), growth control coefficient _{0 }of both R_{c}R_{L }and R_{c}R_{P }dimers equal to 100; on lower plot, division times against generation number for different initial N_{0 }values are displayed. In the upper plots, the vertical dashed lines represent the cell division times that take place when ^{1/3}.

Having determined optimal external conditions, the influence of ribozyme length is now investigated by keeping all the other kinetic parameters constant. Calculations have been performed changing in turn the length of R_{L }or R_{P}, and fixing the size of the other ribozyme to 100 bases, or changing the size of both R_{L }and R_{P }but keeping the same length. Results are reported in Figure _{25}, with a larger radius ρ_{25 }and a lower RNA total concentration [RNA Strands]. The variations observed are quite small compared to those for 100-base ribozymes, except for the [RNA Strands] that show a change about 15%. Furthermore, the Ribocell shows to be much more sensitive to the change in size of the polymerase ribozyme R_{P }rather than R_{L}. This can be ascribed to the fact that, being longer, R_{P}, requires more time to self-replicate and this decreases the overall concentration of all the polymerase ribozymes and in turn the efficiency of genome self-replication and membrane self-reproduction.

Influences of ribozyme length on the Ribocell stationary regime

**Influences of ribozyme length on the Ribocell stationary regime**. Life time Δ_{25}, radius ρ_{25 }and RNA total strand concentration [RNA Strands]_{25 }after 25 generations are reported against ribozyme length. The legend reports the ribozymes that are changed in size.

Finally, Figure _{25 }on the kinetic constants for RNA dimer formation _{SS }and dissociation _{S}. The plot clearly shows that the Ribocell life cycle at stationary regimes does not depend explicitly on the kinetic constant single values _{SS }and _{S }but on their ratio: _{SS}/_{S}, that is on the thermodynamic constant of RNA dimerization. The more thermodynamically stable the RNA dimers, the longer it takes to observe Ribocell self-reproduction. For instance, if _{SS}/_{S }is decreased by two orders of magnitude, the Ribocell life time reduces from 68.2 days to 11.8-6.4 days. The study of Ribocell time behavior approaching the stationary regime as a function of _{SS }and _{S }values would require a much deeper analysis that is beyond the scope of this paper.

Influences of _{ss }and _{s }kinetic constants on the Ribocell stationary regime

**Influences of k _{ss }and k_{s }kinetic constants on the Ribocell stationary regime**. Life time Δ

Stochastic simulations

Stochastic simulations were performed by means of the parallel version of ENVIRONMENT, running 32 statistically equivalent simulations of a 10-ribocell solution on different CPUs. Therefore, the outcomes were obtained as averages from a population of 320 vesicles. Kinetic parameters used for simulations are those reported in Table

On the left in Figure _{c}R_{L }and R_{c}R_{P}. At the end of the simulations of all three cases, similar compositions of the protocell population are obtained with low percentages of real ribocells (3.3-6.7%) while the most populated fractions are those of empty (40.0-41.7%) self-producing (26.7-33.3%) and broken (18.3-25.0%) vesicles, respectively. Reduced ribocells are present only in the first generations since they very soon decay into self-producing and empty vesicles. Inert vesicles, i.e. vesicles entrapping free chains of _{c}R_{P }and/or _{c}R_{L }or a single R_{P}, are not formed and this can be ascribed to the high stability of RNA dimers and complexes so that the chance of finding free RNA monomers at the time of vesicle division is extremely improbable. Thus, the three studied cases are differentiated by their time behavior rather than by the final protocell population, as confirmed by the plots on the right in Figure _{n }> and the number of dividing protocells are reported against the generation number. In agreement with deterministic predictions, for the first generations the average division times <Δ_{n }> are higher for ribocells starting with a lower initial number _{0 }of dimers although, in all cases, the deterministic Δ_{n }(red triangles) are greater than the stochastic averages (black circles with error bars). This can be partially ascribed to the fact that the average <Δ_{n }> is calculated on all the protocells that undergo the _{L }monomer is present and this lowers the average division time.

Stochastic simulation outcomes

**Stochastic simulation outcomes**. Plots on the left show the time evolution of ribocell populations while the legend shows the final compositions; plots on the right report the stochastic <Δt_{n }> (circles with error bars) and deterministic Δt_{n }(red triangles) division times on the left axis while the percentage of dividing protocells (dashed gray lines) is reported on the right axis against the generation number _{c}R_{L }and R_{c}R_{P}.

As an example, Figure _{c}R_{L }and one R_{c}R_{P}. In the upper plot, a comparison between the reduced surface time course determined deterministically and simulated stochastically is shown. As can be seen, the stochastic time trend presents a very irregular time behavior compared to the deterministic one that describes a highly synchronized oscillating regime of growth and division. In contrast, stochastic simulations highlight the alternation of dormant phases, where the reduced surface remains practically constant, both the core volume and the membrane surface being constant (data not shown), to very active steps where protocell growth takes place very fast, leading to a division event.

Stochastic time behavior of a single ribocell with _{0 }= 1

**Stochastic time behavior of a single ribocell with N _{0 }= 1**. In the upper plot, the stochastic time trend (black line) of the reduced surface

In order to account for this behavior, in Figure _{L }and polymerase R_{P }ribozymes present as monomers, complementary strands, dimers and complexes, are reported against time, along with the number of free R_{L }and R_{P }monomers that exhibit catalytic activity. As can be seen, the fast growth and division step corresponds to the presence in the vesicle core of a free R_{L }chain while, in the dormant phase, ribozymes are all coupled in the form of dimers or complexes. In fact, during RNA template transcription in the first generation life time, the volume remains practically constant since the amount of waste molecules produced is not sufficient to promote a substantial water flux from the external environment. As a consequence, self-producing vesicles with a genome made up only of R_{L }monomers can reproduce very efficiently since no dormant phase can occur, given that the formation of R_{c}R_{L }dimers is impossible. This is what happens at high generation numbers in the protocell population time evolutions reported in Figure _{0 }equal to 1 and 10. In fact, at high generation numbers, the only dividing protocells are self-producing vesicles that present free R_{L }monomers in the core volume. Although there are very few of these protocells, they can divide very efficiently, with a Δ_{n }

Stochastic time evolution of the genome composition of a single ribocell with _{0 }= 1

**Stochastic time evolution of the genome composition of a single ribocell with N _{0 }= 1**. In the upper plot, the total number of lipase strands (gray lines) and the R

Conclusions

In this paper, we applied an already published Ribocell _{L}, was set equal to 1.7E+3s^{-1}M^{-1}, the lowest value that exhibited a stationary regime of growth and division in a previous work

By means of deterministic analysis, the robustness of the stationary regime was also investigated as a function of the initial conditions, the length of ribozymes and the kinetic constants of the RNA dimerization. For 100-base long ribozymes, the best experimental conditions in terms of the external concentrations have been found: [_{ex}] = [_{ex}] = 1.0E-2M and [_{ex}] = 0.3M in order to observe a stationary regime with the lowest division time: Δ_{20} = 68.3 days. A so high protocell life time can be mainly ascribed to the RNA reversible association that is shifted towards the dimer formation rendering the concentration of the catalyzers R_{L }and R_{P }very low. This has been confirmed by analyzing the dependence of the life cycle on the thermodynamic constant of ribozyme dimerization. A small influence on the stationary regime was observed on changing the length of the RNA strands: an increase in filament size determines higher life times and a lower amount of genetic material. This effect is more pronounced when the R_{P }length changes. On the other hand, the external concentrations of substrates and inert compounds appear to highly affect the stationary regime in terms of vesicle size, genetic material amount and genome composition. Moreover, deterministic calculations have also shown that the stationary regime can be reached from very different initial genome composition, although when too much free R_{L }ribozymes are present at the beginning the death for ribozymes segregation can be observed since the protocell divides too quickly before the genome replication. Stochastic simulations have been done starting from a population of 320 identical ribocells with an initial genome composed by _{0 }= 1,10 and 100 dimers of both R_{c}R_{L }and R_{c}R_{P}. The analysis of simulations outcomes shows that the ribocell time behaviors is highly influenced by random fluctuations. Since the genetic material is randomly distributed at each cell division, this can produce different type of protocells, ranging from empty vesicles to genuine ribocells, their internal metabolism being highly influenced by the presence of the catalytic RNA strands. In fact, deterministic analysis cannot take into account the disappearance of ribozymes due to a vesicle division, since this approach simply halves the genetic amount and follows the reacting molecule time courses in terms of population averages, i.e. real positive numbers that can be less than one without being zero. On the other hand, the stochastic simulations are more realistic to the random loss of ribozymes from the genome being capable of describing a population of protocells with completely different time behaviors. As a consequence, the simulation outcomes show that ribocells are not enough robust to survive to random fluctuations. In fact only about the 5% of the initial population survive as genuine ribocells after 15-25 generations and on a longer time window they are destined for extinction. Furthermore, the time course of each single protocell is also greatly influenced by intrinsic stochasticity in particular by the time fluctuations of the RNA dimer dissociation. In fact, when all the RNA strands are associated in dimers, protocells remain in a lazy phase, whereas free R_{L }monomers induce fast growth and division steps and free R_{P }cause the fast RNA replication without changing the vesicle size appreciably. Therefore these two process are synchronized only by chance and this also represents a reason of weakness of this model protocell.

In order to implement experimentally ribozymes-based minimal cells two main improvements are necessary. As first, more free monomers of both R_{L }and R_{P }must be available in the vesicle core so that the ribocell life cycle will be speeded up and the division time lowered. This can be achieved by increasing the working temperature since it has been recently show that fatty acid vesicles are stable up to 90°C _{SS }value used can be considered as the appropriate 100-base long RNA association constant for a higher temperature than 25°C _{c}R_{L }dissociation since when free R_{L }monomers are present in the aqueous core the membrane growth quickly and the division takes place very soon. Thus the R_{c}R_{L }dissociation can act as a trigger for the membrane growth and division.

Finally rephrasing the George Box famous sentence, we are aware that this

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

This work has been funded by the MIUR (PRIN2008FY7RJ4_002). Parallel calculations were done at CINECA Super Computing Italian Centre thanks to the approved ISCRAC project (HP10CVJLGZ). The authors would like to thank Anthony Green for his careful review of the English in this manuscript.

This article has been published as part of