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A stochastic model of catalytic reaction networks in protocells

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Abstract

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.

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Notes

  1. Limitations on the outflow can be modeled in a chemostat e.g. by supposing that all the molecules that are larger than a certain size precipitate and cannot be washed away.

  2. It is worthwhile to notice that the presence of the“ catalysis” within the tuple allows the possibility for a species to catalyze more than one reaction and for a reaction to be catalyzed by more than one species.

  3. It has been observed that some superconcentration phenomena can take place under particular circumstances (Pereira de Souza et al. 2012; Serra and Villani 2013) but we will neglect them here.

  4. Exceptionally for the case 1 μM, \(\Theta \) is not computed after 3,000 s but when at least 5,000 reactions have occurred within the simulation. The reason for this is that the low concentrations involve a so slow dynamics that 3,000 s are not enough in order to observe significant chemical changes.

  5. In regard to CH2 we are here considering the value of \(\Theta \) excluding the species belonging to the RAF set (last columns of the table). Since the molecules belonging to the RAF set reach a concentration much greater with respect to the other molecules, considering them in the angle computation would misrepresent the distance among the simulations.

  6. We set the sampling frequency and the time threshold of the windows by taking advantage from several initial model threads, not essential to the comprehension of this article.

  7. This property was proved earlier by Munteanu et al. for the Los Alamos bug mode (Munteanu et al. 2007).

  8. https://github.com/paxelito/carness

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Acknowledgments

Stuart Kauffman, Norman Packard and Wim Hordijk kindly shared with us their deep understanding of autocatalytic sets in several useful discussions. Useful discussions with Ruedi Füchslin, Davide De Lucrezia, Timoteo Carletti, Andrea Roli and Giulio Caravagna are also gratefully acknowledged. The authors are also grateful to Giulia Begal for kindly drawing the image of Fig. 2. C.D. wishes to acknowledge the project SysBionet (12-4-5148000-15; Imp. 611/12; CUP: H41J12000060001; U.A. 53) for the financial support of the work.

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Authors and Affiliations

  1. Department of Physics, Informatics and Mathematics, Modena and Reggio Emilia University, Modena, Italy

    Roberto Serra & Marco Villani

  2. European Centre for Living Technology, ECLT, University Ca’ Foscari of Venice, Venice, Italy

    Roberto Serra, Alessandro Filisetti & Marco Villani

  3. Department of Informatics, Systems and Communication, University of Milan Bicocca, Milan, Italy

    Alex Graudenzi & Chiara Damiani

  4. SYSBIO - Centre for Systems Biology, Piazza della Scienza 2, 20126, Milan, Italy

    Chiara Damiani

  5. Department of Economics and Statistics “Cognetti de Martiis”, University of Turin, Turin, Italy

    Tommaso Panini

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  1. Roberto Serra
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  2. Alessandro Filisetti
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  4. Alex Graudenzi
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Corresponding author

Correspondence to Roberto Serra.

Appendix 1: Simulation environment and parameter settings

Appendix 1: Simulation environment and parameter settings

Simulations were performed with the CaRNeSS simulatorFootnote 8 developed by the research group.

In the following, the baseline setting of the system used in the simulations is reported (for the parameters that were variated in the different experiments please refer to the text): \((\bullet )\) Alphabet: A, B, \((\bullet )\) Volume = \( = 1e-18 dm^3 = 1{\rm \mu} ^3\), \((\bullet )\) Average catalysis probability = 1 catalyzed reaction for species, \((\bullet )\) Maximum length of the species, \(L_{max} = 6\), \((\bullet )\) \(L_{perm} = 2\), \((\bullet )\) Monomers and dimers do NOT catalyze, \((\bullet )\) \(K_{cleav}= 25M^{-1}sec^{-1}\), \((\bullet )\) \(K_{comp}=50M^{-1}sec^{-1}\), \((\bullet )\) \(K_{diss}=1M^{-1}sec^{-1}\), \((\bullet )\) \(K_{cond}=50M^{-1}sec^{-1}\).

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Serra, R., Filisetti, A., Villani, M. et al. A stochastic model of catalytic reaction networks in protocells. Nat Comput 13, 367–377 (2014). https://doi.org/10.1007/s11047-014-9445-6

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