Stationary Distributions for Asymmetrical Autocatalytic Reaction Networks with Discreteness-induced Transitions (DITs)

The phenomenon of discreteness-induced transitions is highly stochastic dependent dynamics observed in a family of autocatalytic chemical reaction networks including the acclaimed Togashi Kaneko model. These reaction networks describe the behaviour of several different species interacting with each other, and the counts of species concentrate in different extreme possible values, occasionally switching between them. This phenomenon is only observed under some regimes of rate parameters in the network, where stochastic effects of small counts of species takes effect. The dynamics for networks in this family is ergodic with a unique stationary distribution. While an analytic expression for the stationary distribution in the special case of symmetric autocatalytic behaviour was derived by Bibbona, Kim, and Wiuf, not much is known about it in the general case. Here we provide a candidate distribution for reaction networks when the autocatalytic rates are different. It was inspired by a model in population genetics, the Moran model with genic selection, which shares many similar reaction dynamics to our autocatalytic networks. We show that this distribution is stationary when autocatalytic rates are equal, and that it is close to stationary when they are not equal

Stationary distributions and condensation in autocatalytic CRN

We investigate a broad family of non weakly reversible stochastically modeled reaction networks (CRN), by looking at their steady-state distributions. Most known results on stationary distributions assume weak reversibility and zero deficiency. We first give explicitly product-form steady-state distributions for a class of non weakly reversible autocatalytic CRN of arbitrary deficiency. Examples of interest in statistical mechanics (inclusion process), life sciences and robotics (collective decision making in ant and robot swarms) are provided. The product-form nature of the steady-state then enables the study of condensation in particle systems that are generalizations of the inclusion process.Comment: 25 pages. Some typos corrected, shortened some part

Switching Dynamics in Reaction Networks Induced by Molecular Discreteness

To study the fluctuations and dynamics in chemical reaction processes, stochastic differential equations based on the rate equation involving chemical concentrations are often adopted. When the number of molecules is very small, however, the discreteness in the number of molecules cannot be neglected since the number of molecules must be an integer. This discreteness can be important in biochemical reactions, where the total number of molecules is not significantly larger than the number of chemical species. To elucidate the effects of such discreteness, we study autocatalytic reaction systems comprising several chemical species through stochastic particle simulations. The generation of novel states is observed; it is caused by the extinction of some molecular species due to the discreteness in their number. We demonstrate that the reaction dynamics are switched by a single molecule, which leads to the reconstruction of the acting network structure. We also show the strong dependence of the chemical concentrations on the system size, which is caused by transitions to discreteness-induced novel states.Comment: 11 pages, 5 figure

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

A model of protocell based on the introduction of a semi-permeable membrane in a stochastic model of catalytic reaction networks

In this work we introduce some preliminary analyses on the role of a semi-permeable membrane in the dynamics of a stochastic model of catalytic reaction sets (CRSs) of molecules. The results of the simulations performed on ensembles of randomly generated reaction schemes highlight remarkable differences between this very simple protocell description model and the classical case of the continuous stirred-tank reactor (CSTR). In particular, in the CSTR case, distinct simulations with the same reaction scheme reach the same dynamical equilibrium, whereas, in the protocell case, simulations with identical reaction schemes can reach very different dynamical states, despite starting from the same initial conditions.Comment: In Proceedings Wivace 2013, arXiv:1309.712

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

Unifying autocatalytic and zeroth order branching models for growing actin networks

The directed polymerization of actin networks is an essential element of many biological processes, including cell migration. Different theoretical models considering the interplay between the underlying processes of polymerization, capping and branching have resulted in conflicting predictions. One of the main reasons for this discrepancy is the assumption of a branching reaction that is either first order (autocatalytic) or zeroth order in the number of existing filaments. Here we introduce a unifying framework from which the two established scenarios emerge as limiting cases for low and high filament number. A smooth transition between the two cases is found at intermediate conditions. We also derive a threshold for the capping rate, above which autocatalytic growth is predicted at sufficiently low filament number. Below the threshold, zeroth order characteristics are predicted to dominate the dynamics of the network for all accessible filament numbers. Together, this allows cells to grow stable actin networks over a large range of different conditions.Comment: revtex, 5 pages, 4 figure