Abstract
In the last two chapters we have shown several interesting results, which will now be brought together in a quite complete (albeit abstract) protocell model. In Chap. 3 we have studied how the presence of genetic memory molecules (GMMs) can affect the growth and fission rate of their lipid container, leading under quite broad assumptions to the important phenomenon of emergent synchronization, i.e. to a condition where protocell fission and duplication of its genetic material take place at the same pace. In that chapter, chemical kinetics has been described with deterministic differential equations (it has also been mentioned that synchronization is somewhat robust even if small fluctuations are considered).
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Notes
- 1.
When we wish to stress this difference, we use the term vesicle for a cell of fixed size, keeping the term protocell for a growing and dividing entity.
- 2.
“chemical composition” means the various chemical species that are present, and the values of their concentrations.
- 3.
In order to introduce change and novelties, the random appearance of new chemicals has been proposed, for example introducing a new self-replicating set of reactions or totally stopping the occurrence of entire blocks of reactions (Vasas et al. 2012).
- 4.
We always assume that the volume of the “environment” is much larger than that of the protocell, or even of the whole population of protocells, so that the changes in the composition of the environment due to outflows from the protocells are negligible.
- 5.
Pioneered by Gánti with his Chemoton model (Gánti 2003).
- 6.
Due to the observed robustness of emergent synchronization in different models, we hypothesize that the main qualitative results are likely to hold in the case of even division, also if different shapes are assumed.
- 7.
As already commented in Sect. 4.5, a RAF is a set of reactions, but for simplicity, when no misunderstandings are possible, the term RAF will be used to indicate also the set of chemical species involved in the RAF structure.
- 8.
Except for the case of so-called near-membrane reaction models, where the key reactions take place in a thin spherical shell close to the inner side of the membrane; also in this case the concentrations are however homogeneous inside each partition of the total inner volume.
- 9.
In the cases considered here the difference in chemical potential is due only to differences in concentrations.
- 10.
Because of their affinity with the membrane we assume that also these materials can cross it. The consequence of releasing this assumption will be briefly discussed at the end of this chapter.
- 11.
The hypothesis that the two daughters have identical volumes is a non-essential assumption, since the division phenomenon is supposed to happen at a given threshold, independently of the initial size: conversely, it allows a more compact result presentation—see Chap. 6 for a more detailed discussion.
- 12.
The observations of this section are valid both for cells and protocells; for this reason we will sometimes use here the generic term “cell”.
- 13.
Apparently, this is a feature of life as we know it: we never observed living entities as the intelligent cloud wondering among stars depicted in “The Black Cloud” by astrophysicist Fred Hoyle (Hoyle 1957). On the contrary, the organization of all known living entities is based on small units, whose chemical compositions significantly differ from the environmental one.
- 14.
As we will discuss below, a finite size does not guarantee per se the existence of differences among the internal and the external chemical concentrations.
- 15.
Note however that, in order to interpret some recent experimental data (Souza et al. 2009), it has been suggested that some processes might take place, when the membrane closes, that favor the onset of some concentration differences between inside and outside.
- 16.
We consider the case of increased reaction rate, but the same reasoning could be applied, mutatis mutandis, to the case where the membrane slows down some reaction.
- 17.
Unless of course it is consumed by another reaction, but the reasoning above suffices to show that the concentrations can easily become different.
- 18.
Note that in this case the CSTR is a macroscopic device that is the (open) environment where the protocell lives, while the model of the protocell is that of a semipermeable vesicle with finite transmembrane diffusion rates.
- 19.
This is a perhaps unexpected phenomenon: starting from a seemingly equilibrium situation where the internal and external concentrations are equal to each other, a (transient) difference appears. The explanation lies in the fact that the initial condition of vanishing concentrations in case (a) of Fig. 5.4 is an equilibrium when no reaction A↔X is taking place, but it is out of equilibrium when the reaction is “turned on” in the simulated system.
- 20.
That, under the above assumptions, are equal to those of an “equivalent volume”.
- 21.
Let us recall from Chap. 4 the notion of a “chemistry”, i.e. a set of tuples {species; catalyzes; reactions}, where the species catalyzes the reaction. In order to understand generic behaviors, we analyze different chemistries.
- 22.
The differences between these two chemistries are representative of those observed in the larger sample.
- 23.
This is why we use angles (Eq. 5.3) to measure the differences between compositions, instead of Euclidean distances.
- 24.
We always extracted the number of molecules for each chemical species from a Gaussian distribution respectively 0.01 mM and 1 µM on average.
- 25.
In this chapter we are neglecting cases where synchronization occurs in obvious ways, like, for example the situation where a molecular type in the food set (i) directly contributes to the growth of the container, (ii) catalyzes the condensation of a chemical species that is not substrate of any reaction in the given chemistry, but directly contributes to the container growth. These situations (easily tractable with the techniques used in Chap. 3) indeed assure the continuous production of the chemicals coupled to the container and lead to synchronization.
- 26.
As anticipated, in this chapter we suppose that no food molecule is a catalyst: as a consequence RAF sets need to include a Strongly Connected Component (SCC).
- 27.
- 28.
- 29.
“Density of catalysis” is used here as a shorthand for the probability p that a randomly chosen chemical species catalyses a randomly chosen reaction.
- 30.
In order to better define this concept, a sRAF is the part of a RAF whose chemical species duplicates at the same rate of the container: typically, but not always, this part coincides with the whole RAF. This synchronization property holds as long as there is a single sRAF within the vesicle. As we will see in the following sections, sRAFs having different growth rates sometimes do not coexist within the same container: also in this case we continue to call them “sRAFs”, remembering that the same structures, if they were alone, would be able to sustain the protocell synchronization.
- 31.
Note that protocells in Vasas et al. (2012) are simulated as very small CSTRs.
- 32.
The results summarized below are the outcome of several simulations performed on the model described in Sect. 5.1, assuming instantaneous transmembrane diffusion.
- 33.
If not differently indicated in the following, we consider the same value for all the kinetic constants of the reactions. If this situation does not occur, the chemical groups connected with the higher global growth rate can force the protocell growth and duplication to such a high rate that, generation after generation, the other chemical species dilute and disappear (Villani et al. 2016). This situation does not hold if the hypothesis of fast transmembrane diffusion is released (as we discuss in the final part of this chapter).
- 34.
In a simple “activation energy” model, it may happen that few “outlier” reactants have enough energy to cross the barrier while the average energy does not suffice to do so.
- 35.
The case of equal growth rates (allowing the coexistence of different RAFs) is indeed very rare, and it can be neglected.
- 36.
It is well-known that in case of exponential growth only the fastest competitors can survive, see also Chap. 3.
- 37.
We suppose that sometimes the protocell can maintain both RAFs—this would be hard in the case of significant material losses during cell division but, as it has been said before, we rather consider the case where no such loss occurs.
- 38.
In this case, coexistence of RAFs with difference replication rates is possible even when one takes into account possible losses of chemicals during fission.
- 39.
In such vision the finite diffusion rate through the membrane is one of the key processes that allow the co-existence and the coordination within the same protocell of otherwise chemically independent reactions. Simultaneously there could be other coordination phenomena, as for example the “osmotic coupling” discussed in (Shirt-Ediss et al. 2015).
- 40.
While the simulations refer to the RAFs of Fig. 5.13, their general properties are common to several other cases that have been examined and therefore provide some indications on how advantageous novelties can indeed develop within the system.
- 41.
The range of parameters allowing these behaviors is wider in the case of reactions occurring inside the whole system’s volume (Villani et al. 2016, Calvanese et al. 2017).
- 42.
Evolvability is the ability of a population to not merely generate diversity, but to generate adaptive diversity, and thereby evolve through natural (or artificial) selection.
- 43.
Obviously, both protocells and living cells are dynamical objects, and even the simple translation of information in dynamical behaviors is per se a complex dynamical process. Moreover, during the splitting processes the parent cells transmit to the descendants—beside the static information—also their dynamical state.
- 44.
- 45.
Different reactions could have different K cl values: in order to avoid an excessive pedantry in Eq. 5.13 we omitted the subscript r.
- 46.
Different reactions could have different constant values: in order to avoid an excessive pedantry in Eq. 5.14 we omitted the subscript d.
- 47.
That is, we need several Δtg intervals in order to cover a single time step of the Euler framework. This fact allows the synchronization between the Gillespie and the Euler framework proposed by steps (i)–(iii).
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Serra, R., Villani, M. (2017). A Stochastic Model of Growing and Dividing Protocells. In: Modelling Protocells. Understanding Complex Systems. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1160-7_5
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