Roberto Serra’s research while affiliated with University of Amsterdam and other places

Random Boolean Networks are dissipative dynamical models of gene regulatory networks, which are older than fifty years but still raise considerable interest. In this paper we will rely on two key concepts which had been introduced in previous works, namely those of pseudo-attractors (which are obtained by projecting true dynamical attractors onto constant vectors) and of the “common sea” (de-fined as the set of nodes which take the same value in every pseudo-attractor of a given network realization). In particular, we will study the dependence of the number of pseudo-attractors and of the relative size of the common sea upon the values of some key parameters, like the average number of connections per node and the so-called bias of the set of Boolean functions, paying particular attention to dynamically critical networks. We will also comment on the relationship of these models with measured gene expression values in single-cell observations.

The conditions that allow for the sustained growth of a protocell population are investigated in the case of asymmetrical division. The results are compared to those of previous studies concerning models of symmetrical division, where synchronization (between duplication of the genetic material and fission of the lipid container) was found under a variety of different assumptions about the kinetic equations and about the place where molecular replication takes place. Such synchronization allows a sustained proliferation of the protocell population. In the asymmetrical case, there can be no true synchronization, since the time to duplication may depend upon the initial size, but we introduce a notion of homogeneous growth that actually allows for the sustained reproduction of a population of protocells. We first analyze Surface Reaction Models, defined in the text, and we show that in many cases they undergo homogeneous growth under the same kinetic laws that lead to synchronization in the symmetrical case. This is the case also for Internal Reaction Models (IRMs), which, however, require a deeper understanding of what homogeneous growth actually means, as discussed below.

The conditions which allow sustained growth of a protocell population are investigated in the case of asymmetrical division. The results are compared to those of previous studies concerning models of symmetrical division, where synchronization (between duplication of the genetic material and fission of the lipid container) was found under a variety of different assumptions about the kinetic equations and about the place where molecular replication takes place. Such synchronization allows a sustained proliferation of the protocell population. In the asymmetrical case there can be no true synchronization, since the time to duplication may depend upon the initial size, but we introduce a notion of homogeneous growth which actually allows sustained reproduction of a population of protocells. We first analyze Surface Reaction Models, defined in the text, and we show that in many cases they undergo homogeneous growth under the same kinetic laws which lead to synchronization in the symmetrical case. This is the case also for Internal Reaction Models (IRMs), which however require a deeper understanding of what homogeneous growth actually means, as discussed below.

A key question concerning the origin of life is whether polymers, such as nucleic acids and proteins, can spontaneously form in prebiotic conditions. Several studies have shown that, by alternating (i) a phase in which a system is in a water-rich condition and (ii) one in which there is a relatively small amount of water, it is possible to achieve polymerization. It can be argued that such “wet-dry” cycles might have actually taken place in the primordial Earth, for example in volcanic lakes. In this paper, using a version of the binary polymer model without catalysis, we have simulated wet and dry cycles to determine the effectiveness of polymerization under these conditions. By observing the behavior of some key variables (e.g., the number of different chemical species which appeared at least once and the maximum length of the species currently present in the system) it is possible to see that the alternation of wet and dry conditions can indeed allow a wider exploration of different chemical species when compared to constant conditions.KeywordsProtocellsOrigin of lifeGillespie algorithmSemipermeable membrane

In complex systems that host evolutionary processes, in which entirely new entities may enter the scene, some variables can sometimes show a “hockey-stick” behavior, that is a long period of slow growth followed by an “explosive” increase. The TAP equation was proposed with the aim of describing the growth of the number of different types of entities in systems where new entities (e.g., artifacts) can be created, supposing that they derive from transformations of pre-existing ones. It shows a very interesting divergence in finite times, different from the usual exponential growth where divergence takes place in the infinite time limit. The TAP equation does not deal with the growth of the number of actual types, but rather with the number of the possible ones (the members of the so-called set of Adjacent Possible), and it can therefore overestimate the actual rate of growth. In this paper, we introduce a model (called BPSM, focused on systems that may be relevant for the origin of life) that takes into account the difference between the Adjacent Possible and the set of types that are actually created. Using simulations, it has been observed that the growth of the number of chemical species in the system resembles that of the corresponding TAP equation. Since in this case only combinations of at most two entities can be considered at each time, the TAP equation can be analytically integrated. Its behavior can be then compared to the (necessarily finite) behavior of model simulations; their behaviors turn out to be quite similar, and proper tests are introduced, which show that they differ from the familiar exponential growth. Therefore, the BPSM model provides a description of the rapid increase of diversity which resembles TAP, while based upon the growth of the actual entities rather than on the Adjacent Possible.KeywordsInnovationBinary polymer modelOrigin of lifeAdjacent possibleTAP equation

The principle of dynamical criticality is a very important hypotheses in biology, and it therefore deserves a thorough investigation. Testing the principle in real biological cases can be far from trivial: therefore, in this work we make use of the Random Boolean Network framework, which has been extensively used to model genetic regulatory networks, and which has since become one of the most used models in the field of complex systems. We subject several RBN ensembles to evolutionary changes: the key research questions are whether initially critical networks will grow faster than ordered or chaotic ones, and whether evolution can influence the dynamic regime, and in which direction. The results obtained so far indicate that critical systems perform well in the analyzed tasks. In the case of two connections per node, the best performances are those of critical systems, while increasing the value of the connectivity there seems to be a slight shift towards more disordered regimes (albeit still close to the critical one).KeywordsRBNEvolutionCritical dynamic regime

The Theory of the Adjacent Possible (TAP) equation has been proposed as an appropriate description of super-exponential growth phenomena, where a phase of slow growth is followed by a rapid increase, leading to a “hockey stick” curve. This equation, initially conceived to describe the growth in time of the number of new types of artifacts, has also been applied to several natural phenomena. A possible drawback is that it may overestimate the number of new artifact types, since it does not take into account the fact that interactions, among existing types, may produce types which have already been previously discovered. We introduce here a Binary String World (BSW) where new string types can be generated by interactions among (at most two) already existing types. We introduce a continuous limit of the TAP equation for the BSW; we solve it analytically and show that it leads to divergence in finite time. We also introduce a criterion to distinguish this type of behavior from the familiar exponential growth, which diverges only as t ® µ. In the BSW, it is possible to directly model the generation of new types, and to check whether the newborns are actually novel types, thus discarding the rediscoveries of already existing types. We show that the type of growth is still TAP-like, rather than exponential, although of course in simulations one never can observes true divergence. We also show that this property is robust with respect to some changes in the model, as long as it deals with types (and not with individuals).

Random Boolean Networks (RBNs for short) are strongly simplified models of gene regulatory networks (GRNs), which have also been widely studied as abstract models of complex systems and have been used to simulate different phenomena. We define the “common sea” (CS) as the set of nodes that take the same value in all the attractors of a given network realization, and the “specific part” (SP) as the set of all the other nodes, and we study their properties in different ensembles, generated with different parameter values. Both the CS and of the SP can be composed of one or more weakly connected components, which are emergent intermediate-level structures. We show that the study of these sets provides very important information about the behavior of the model. The distribution of distances between attractors is also examined. Moreover, we show how the notion of a “common sea” of genes can be used to analyze data from single-cell experiments.

It is argued that, in order to understand how life might have been originated under abiotic conditions, it is necessary to describe the appearance of both primitive cells and self-replicating sets of molecules. It is observed that, under quite general assumptions, the fluctuations in the internal composition of small protocells may be an important factor, which allows self-replication to take place by limiting the number of different types of molecular species and of reactions. The notion of a shadow biosphere is also briefly discussed.