The stochastic evolution of a protocell. The Gillespie algorithm in a dynamically varying volume

📝 Abstract

In the present paper we propose an improvement of the Gillespie algorithm allowing us to study the time evolution of an ensemble of chemical reactions occurring in a varying volume, whose growth is directly related to the amount of some specific molecules, belonging to the reactions set. This allows us to study the stochastic evolution of a protocell, whose volume increases because of the production of container molecules. Several protocells models are considered and compared with the deterministic models.

💡 Analysis

In the present paper we propose an improvement of the Gillespie algorithm allowing us to study the time evolution of an ensemble of chemical reactions occurring in a varying volume, whose growth is directly related to the amount of some specific molecules, belonging to the reactions set. This allows us to study the stochastic evolution of a protocell, whose volume increases because of the production of container molecules. Several protocells models are considered and compared with the deterministic models.

📄 Content

arXiv:1112.1281v1 [physics.bio-ph] 6 Dec 2011 The stochastic evolution of a protocell. The Gillespie algorithm in a dynamically varying volume. by T. Carletti and A. Filisetti Report naXys-22-2011 6 12 2012 Namur Center for Complex Systems University of Namur 8, rempart de la vierge, B5000 Namur (Belgium) http://www.naxys.be THE STOCHASTIC EVOLUTION OF A PROTOCELL. THE GILLESPIE ALGORITHM IN A DYNAMICALLY VARYING VOLUME. T. CARLETTI AND A. FILISETTI Abstract. In the present paper we propose an improvement of the Gillespie algorithm allowing us to study the time evolution of an ensemble of chemical reactions occurring in a varying volume, whose growth is directly related to the amount of some specific molecules, belonging to the reactions set. This allows us to study the stochastic evolution of a protocell, whose volume increases because of the production of container molecules. Several protocells models are considered and compared with the deterministic models.

1. Introduction All known life forms are composed of basic units called cells; this holds true from the single-cell prokaryote bacterium to the highly sophisticated eucaryotes, whose existence is the result of the coordination, in term of self-organization and emergence, of the behavior of each single basic unit. While present day cells are endowed with highly sophisticated regulatory mechanisms, that represent the outcome of almost four billion-years of evolution, it is generally believed that the first life-forms were much simpler. Such primordial life-bricks, the protocells, were most probably exhibiting only few simplified functionalities, that required a primitive embodiment structure, a protometabolism and a rudimentary genetics, so to guarantee that offsprings were “similar” to their parents [1, 15, 17]. Intense research programs are being established aiming at obtaining protocells capable of growth and duplication, endowed with some limited form of genetics [12, 13, 14, 17]. Despite all efforts, artificial protocells have not yet been reproduced in laboratory and it is thus extremely impor- tant to develop reference models [3, 10, 14, 16] that capture the essence of the first protocells appeared on Earth and enable to monitor their subsequent evolution. Due to the uncertainties about the details, high-level abstract models are particularly relevant. Quoting Kaneko [7] it is necessary to consider “simplified models able to capture universal behaviors, without carefully adding complicating details”. Most of the models present in the literature are based on deterministic differential equations governing the evolution of the concentrations of the involved reacting molecules. Even if the results are worth discussing and provide important insights, it should be stressed that the former assumptions are rarely satisfied in a cell [5]. Firstly, the number of involved molecules is small and should be counted by integer numbers, hence the use of the concentrations can be questioned; secondly, the presence of the thermal noise introduces in the system a degree of stochasticity than cannot be trivially encoded by a differential equation, mostly because this makes the time evolution a stochastic process. One possible way to overcome such difficulties is to use the Chemical Master equation: given the present state of the system, namely the number of available molecules for each species, and the possible reactions among them, one can compute the transition probabilities to reach and leave the given state and thus get a partial differential equation describing the time evolution of the probability distribution of having a given number of molecules at any future times [5, 6]. Analytically solving the resulting equation is normally a very hard task, one should thus resort to use numerical methods. A particularly suitable one is the algorithm presented by Gillespie [5, 6], allowing to determine, as a function of the present state of the system, the most probable reaction and the most probable reaction time, i.e. the time at which such reaction will occur. Date: September 14, 2018. 1 2 T. CARLETTI AND A. FILISETTI Let us however observe that in the setting we are hereby interested in, the chemical reactions occur in a varying volume, because of the protocell growth; we thus need to adapt the Gillespie method to account for this factor. To the best of our knowledge, there are in the literature very few papers dealing with the Gillespie algorithm in a varying volume [8, 9]. Moreover in all these papers, the volume variation can be considered as an exogenous factor, not being directly related to the number of lipids forming the protocell membrane. So our main contribution is to improve the Gillespie algorithm taking into account the protocell varying volume which is moreover consistent with the increase of the number of lipids constituting the protocell membrane. The paper is organized as follow. In Section 2 we briefly recall the Surface Reaction Models of protocell, that would

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