The sustained growth of a population of protocells which undergo symmetrical division (where each individual splits into two equal daughter protocells) requires synchronization between the two processes of (i) duplication of the genetic material and (ii) fission of the lipid container. It has however been observed that one often encounters uneven division, where daughters of different sizes may be generated. Here we analyze the case of asymmetrical division, where each protocell has exactly two daughters of different sizes. In this case no true synchronization is possible, and we introduce the notion of homogeneous growth which guarantees that sustained population growth is possible. We consider different abstract models of protocells growth and reproduction and we show by simulation that homogeneous growth is encountered, both in Surface Reaction Models, where the replicators are located in the membrane, and in Internal Reaction Models where they are found in the internal water phase, under a broad set of different kinetic equations. We argue that, when there are different kinds of replicators, it is legitimate to identify the “chemical signature” of the protocell with the set of the ratios between the quantities of these replicators at fission time: it is shown that, in the case of linear kinetic equations, the ratios and therefore the chemical identity are conserved through generations.

In this extended abstract two novel concepts are defined in the study of Random Boolean Networks, i.e. those of “pseudoattractors” and “common sea”, and it is shown how their analogues can be measured in experimental data on gene expression in single cells.

Modern biological cells are endowed with effective mechanisms which control their division, ensuring that it does not take place before the duplication of the genetic material has been completed. It is unlikely that similar sophisticated mechanisms were in place in primitive protocells, which were much simpler than their present-day descendants. So a major question concerns the way in which reproduction of the whole protocell might take place together with replication of its genetic molecules, absent any kind of high-level control. This might happen if the rate of duplication of the genetic material and that of fission of the protocell are the same, i.e. if the two processes are synchronized. This possibility can be studied using simplified models of reaction networks (among replicators), assuming that one or more replicators can affect the growth and fission rates of their lipid container. Surprisingly enough, such synchronization does not necessarily require a careful assembly of reactions with very specific reaction rates. On the contrary, it turns out to be a property which emerges spontaneously in a broad set of models, with different parameters, different reaction networks and even different protocell architectures. Note that synchronization, while being a widespread property, is not always achieved for all the models and reaction types. The conditions for emergent synchronization will be discussed, reviewing previous work and showing some new results. These results are based upon dynamical models which assume that the reactions are known a priori. On the other hand, in models of the origin of life it is often assumed that not all the important chemicals are there since the very beginning, but that some of them are synthesized at later stages. The appearance of new chemicals makes new reactions possible, which may in turn lead to the synthesis of new chemicals, etc. Dealing with this kind of problems requires the choice of a particular model of the replicators and of their interactions; in this paper the random binary polymer model proposed by S. Kauffman, where the replicators are polymers which can undergo cleavage or condensation, will be considered. This model allows, in principle, the appearance in time of polymers of increasing length. Another aspect which has to be taken into account, in order to properly model these phenomena, is that new chemical species may be initially present in very low concentrations, which require a stochastic treatment like the one allowed by the well-known Gillespie algorithm. The random binary polymer model can give rise in time to collectively autocatalytic sets, which are able to self-replicate; if some chemicals which belong to the core or to the periphery of these sets are coupled to the growth of the lipid container, this may lead to emergent synchronization. However, the interactions can be quite complicated and the overall behaviour can be counterintuitive. Some examples of dynamical behaviours which have been observed in simulations will be presented and discussed, with particular emphasis on features which are always, or frequently, observed. It will be argued that studying the dynamical interaction of autocatalytic sets with the growth and splitting dynamics of the lipid container is crucial to understand the possibility that a population of protocells undergo sustainable growth and evolution.

Cellular types of multicellular organisms are the stable results of complex intertwined processes that occur in biological cells. Among the many others, chromatin dynamics significantly contributes—by modulating access to genes—to differential gene expression, and ultimately to determine cell types. Here, we propose a dynamical model of differentiation based on a simplified bio-inspired methylation mechanism in Boolean models of GRNs. Preliminary results show that, as the number of methylated nodes increases, there is a decrease in attractor number and networks tend to assume dynamical behaviours typical of ordered ensembles. At the same time, results show that this mechanism does not affect the possibility of generating path dependent differentiation: cell types determined by the specific sequence of methylated genes.

The detection of critical states is a task of utmost importance in complex systems; to this aim, measures to identify such conditions are required. In general, the term criticality concerns the existence of two qualitatively different behaviours that a system can exhibit, which depends on some parameter values. In this short communication, we summarise our recent findings on the use of the Relevance Index to identify critical states in complex systems. Although the Relevance Index method was originally developed to identify relevant sets of variables in dynamical systems, we show that it is also able to detect features of criticality. The index is applied to two notable examples showing slightly different meanings of criticality, namely, the Ising model and Random Boolean Networks. Results show that this index is maximised at critical states and is robust with respect to system size and sampling effort.