A stochastic model of catalytic reaction networks in protocells

Protocells are supposed to have played a key role in the self-organizing processes leading to the emergence of life. Existing models either (i) describe protocell architecture and dynamics, given the existence of sets of collectively self-replicating molecules for granted, or (ii) describe the emergence of the aforementioned sets from an ensemble of random molecules in a simple experimental setting (e.g. a closed system or a steady-state flow reactor) that does not properly describe a protocell. In this paper we present a model that goes beyond these limitations by describing the dynamics of sets of replicating molecules within a lipid vesicle. We adopt the simplest possible protocell architecture, by considering a semi-permeable membrane that selects the molecular types that are allowed to enter or exit the protocell and by assuming that the reactions take place in the aqueous phase in the internal compartment. As a first approximation, we ignore the protocell growth and division dynamics. The behavior of catalytic reaction networks is then simulated by means of a stochastic model that accounts for the creation and the extinction of species and reactions. While this is not yet an exhaustive protocell model, it already provides clues regarding some processes that are relevant for understanding the conditions that can enable a population of protocells to undergo evolution and selection.Comment: 20 pages, 5 figure

Autocatalytic sets in a partitioned biochemical network

In previous work, RAF theory has been developed as a tool for making theoretical progress on the origin of life question, providing insight into the structure and occurrence of self-sustaining and collectively autocatalytic sets within catalytic polymer networks. We present here an extension in which there are two "independent" polymer sets, where catalysis occurs within and between the sets, but there are no reactions combining polymers from both sets. Such an extension reflects the interaction between nucleic acids and peptides observed in modern cells and proposed forms of early life.Comment: 28 pages, 8 figure

On RAF Sets and Autocatalytic Cycles in Random Reaction Networks

The emergence of autocatalytic sets of molecules seems to have played an important role in the origin of life context. Although the possibility to reproduce this emergence in laboratory has received considerable attention, this is still far from being achieved. In order to unravel some key properties enabling the emergence of structures potentially able to sustain their own existence and growth, in this work we investigate the probability to observe them in ensembles of random catalytic reaction networks characterized by different structural properties. From the point of view of network topology, an autocatalytic set have been defined either in term of strongly connected components (SCCs) or as reflexively autocatalytic and food-generated sets (RAFs). We observe that the average level of catalysis differently affects the probability to observe a SCC or a RAF, highlighting the existence of a region where the former can be observed, whereas the latter cannot. This parameter also affects the composition of the RAF, which can be further characterized into linear structures, autocatalysis or SCCs. Interestingly, we show that the different network topology (uniform as opposed to power-law catalysis systems) does not have a significantly divergent impact on SCCs and RAFs appearance, whereas the proportion between cleavages and condensations seems instead to play a role. A major factor that limits the probability of RAF appearance and that may explain some of the difficulties encountered in laboratory seems to be the presence of molecules which can accumulate without being substrate or catalyst of any reaction.Comment: pp 113-12

An algebraic characterization of self-generating chemical reaction networks using semigroup models

The ability of a chemical reaction network to generate itself by catalyzed reactions from constantly present environmental food sources is considered a fundamental property in origin-of-life research. Based on Kaufmann's autocatalytic sets, Hordijk and Steel have constructed the versatile formalism of catalytic reaction systems (CRS) to model and to analyze such self-generating networks, which they named reflexively autocatalytic and food generated (RAF). Previously, it was established that the subsequent and simultaenous catalytic functions of the chemicals of a CRS give rise to an algebraic structure, termed a semigroup model. The semigroup model allows to naturally consider the function of any subset of chemicals on the whole CRS. This gives rise to a generative dynamics by iteratively applying the function of a subset to the externally supplied food set. The fixed point of this dynamics yields the maximal self-generating set of chemicals. Moreover, the lattice of all functionally closed self-generating sets of chemicals is discussed and a structure theorem for this lattice is proven. It is also shown that a CRS which contains self-generating sets of chemicals cannot be nilpotent and thus a useful link to the combinatorial theory of finite semigroups is established. The main technical tool introduced and utilized in this work is the representation of the semigroup elements as decorated rooted trees, allowing to translate the generation of chemicals from a given set of resources into the semigroup language.Comment: 33 pages, 6 figure

Sustainable growth and synchronization in protocell models

The growth of a population of protocells requires that the two key processes of replication of the protogenetic material and reproduction of the whole protocell take place at the same rate. While in many ODE-based models such synchronization spontaneously develops, this does not happen in the important case of quadratic growth terms. Here we show that spontaneous synchronization can be recovered (i) by requiring that the transmembrane diffusion of precursors takes place at a finite rate, or (ii) by introducing a finite lifetime of the molecular complexes. We then consider reaction networks that grow by the addition of newly synthesized chemicals in a binary polymer model, and analyze their behaviors in growing and dividing protocells, thereby confirming the importance of (i) and (ii) for synchronization. We describe some interesting phenomena (like long-term oscillations of duplication times) and show that the presence of food-generated autocatalytic cycles is not sufficient to guarantee synchronization: in the case of cycles with a complex structure, it is often observed that only some subcycles survive and synchronize, while others die out. This shows the importance of truly dynamic models that can uncover effects that cannot be detected by static graph theoretical analyses