A stochastic model of autocatalytic reaction networks

Super-Exponential Growth in Models of a Binary String World

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 → ∝. 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).

Asymptotic Information-Theoretic Detection of Dynamical Organization in Complex Systems

The identification of emergent structures in complex dynamical systems is a formidable challenge. We propose a computationally efficient methodology to address such a challenge, based on modeling the state of the system as a set of random variables. Specifically, we present a sieving algorithm to navigate the huge space of all subsets of variables and compare them in terms of a simple index that can be computed without resorting to simulations. We obtain such a simple index by studying the asymptotic distribution of an information-theoretic measure of coordination among variables, when there is no coordination at all, which allows us to fairly compare subsets of variables having different cardinalities. We show that increasing the number of observations allows the identification of larger and larger subsets. As an example of relevant application, we make use of a paradigmatic case regarding the identification of groups in autocatalytic sets of reactions, a chemical situation related to the origin of life problem.

On the Use of Topological Features of Metabolic Networks for the Classification of Cancer Samples

BACKGROUND

The increasing availability of omics data collected from patients affected by severe pathologies, such as cancer, is fostering the development of data science methods for their analysis.

INTRODUCTION

The combination of data integration and machine learning approaches can provide new powerful instruments to tackle the complexity of cancer development and deliver effective diagnostic and prognostic strategies.

METHODS

We explore the possibility of exploiting the topological properties of sample-specific metabolic networks as features in a supervised classification task. Such networks are obtained by projecting transcriptomic data from RNA-seq experiments on genome-wide metabolic models to define weighted networks modeling the overall metabolic activity of a given sample.

RESULTS

We show the classification results on a labeled breast cancer dataset from the TCGA database, including 210 samples (cancer vs. normal). In particular, we investigate how the performance is affected by a threshold-based pruning of the networks by comparing Artificial Neural Networks, Support Vector Machines and Random Forests. Interestingly, the best classification performance is achieved within a small threshold range for all methods, suggesting that it might represent an effective choice to recover useful information while filtering out noise from data. Overall, the best accuracy is achieved with SVMs, which exhibit performances similar to those obtained when gene expression profiles are used as features.

CONCLUSION

These findings demonstrate that the topological properties of sample-specific metabolic networks are effective in classifying cancer and normal samples, suggesting that useful information can be extracted from a relatively limited number of features.

Side Reactions Do Not Completely Disrupt Linear Self-Replicating Chemical Reaction Systems

A crucial question within the fields of origins of life and metabolic networks is whether or not a self-replicating chemical reaction system is able to persist in the presence of side reactions. Due to the strong nonlinear effects involved in such systems, they are often difficult to study analytically. There are however certain conditions that allow for a wide range of these reaction systems to be well described by a set of linear ordinary differential equations. In this article, we elucidate these conditions and present a method to construct and solve such equations. For those linear self-replicating systems, we quantitatively find that the growth rate of the system is simply proportional to the sum of all the rate constants of the reactions that constitute the system (but is nontrivially determined by the relative values). We also give quantitative descriptions of how strongly side reactions need to be coupled with the system in order to completely disrupt the system.

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.

Predicting species emergence in simulated complex pre-biotic networks

An intriguing question in evolution is what would happen if one could "replay" life's tape. Here, we explore the following hypothesis: when replaying the tape, the details ("decorations") of the outcomes would vary but certain "invariants" might emerge across different life-tapes sharing similar initial conditions. We use large-scale simulations of an in silico model of pre-biotic evolution called GARD (Graded Autocatalysis Replication Domain) to test this hypothesis. GARD models the temporal evolution of molecular assemblies, governed by a rates matrix (i.e. network) that biases different molecules' likelihood of joining or leaving a dynamically growing and splitting assembly. Previous studies have shown the emergence of so called compotypes, i.e., species capable of replication and selection response. Here, we apply networks' science to ascertain the degree to which invariants emerge across different life-tapes under GARD dynamics and whether one can predict these invariant from the chemistry specification alone (i.e. GARD's rates network representing initial conditions). We analysed the (complex) rates' network communities and asked whether communities are related (and how) to the emerging species under GARD's dynamic, and found that the communities correspond to the species emerging from the simulations. Importantly, we show how to use the set of communities detected to predict species emergence without performing any simulations. The analysis developed here may impact complex systems simulations in general.

Sequence selection by dynamical symmetry breaking in an autocatalytic binary polymer model

Template-directed replication of nucleic acids is at the essence of all living beings and a major milestone for any origin of life scenario. We present an idealized model of prebiotic sequence replication, where binary polymers act as templates for their autocatalytic replication, thereby serving as each others reactants and products in an intertwined molecular ecology. Our model demonstrates how autocatalysis alters the qualitative and quantitative system dynamics in counterintuitive ways. Most notably, numerical simulations reveal a very strong intrinsic selection mechanism that favors the appearance of a few population structures with highly ordered and repetitive sequence patterns when starting from a pool of monomers. We demonstrate both analytically and through simulation how this "selection of the dullest" is caused by continued symmetry breaking through random fluctuations in the transient dynamics that are amplified by autocatalysis and eventually propagate to the population level. The impact of these observations on related prebiotic mathematical models is discussed.

The Search for Candidate Relevant Subsets of Variables in Complex Systems

We describe a method to identify relevant subsets of variables, useful to understand the organization of a dynamical system. The variables belonging to a relevant subset should have a strong integration with the other variables of the same relevant subset, and a much weaker interaction with the other system variables. On this basis, extending previous work on neural networks, an information-theoretic measure, the dynamical cluster index, is introduced in order to identify good candidate relevant subsets. The method does not require any previous knowledge of the relationships among the system variables, but relies on observations of their values over time. We show its usefulness in several application domains, including: (i) random Boolean networks, where the whole network is made of different subnetworks with different topological relationships (independent or interacting subnetworks); (ii) leader-follower dynamics, subject to noise and fluctuations; (iii) catalytic reaction networks in a flow reactor; (iv) the MAPK signaling pathway in eukaryotes. The validity of the method has been tested in cases where the data are generated by a known dynamical model and the dynamical cluster index is applied in order to uncover significant aspects of its organization; however, it is important that it can also be applied to time series coming from field data without any reference to a model. Given that it is based on relative frequencies of sets of values, the method could be applied also to cases where the data are not ordered in time. Several indications to improve the scope and effectiveness of the dynamical cluster index to analyze the organization of complex systems are finally given.

Growth and division in a dynamic protocell model

In this paper a new model of growing and dividing protocells is described, whose main features are (i) a lipid container that grows according to the composition of the molecular milieu (ii) a set of “genetic memory molecules” (GMMs) that undergo catalytic reactions in the internal aqueous phase and (iii) a set of stochastic kinetic equations for the GMMs. The mass exchange between the external environment and the internal phase is described by simulating a semipermeable membrane and a flow driven by the differences in chemical potentials, thereby avoiding to resort to sometimes misleading simplifications, e.g., that of a flow reactor. Under simple assumptions, it is shown that synchronization takes place between the rate of replication of the GMMs and that of the container, provided that the set of reactions hosts a so-called RAF (Reflexive Autocatalytic, Food-generated) set whose influence on synchronization is hereafter discussed. It is also shown that a slight modification of the basic model that takes into account a rate-limiting term, makes possible the growth of novelties, allowing in such a way suitable evolution: so the model represents an effective basis for understanding the main abstract properties of populations of protocells.