## Abstract

The goal of synthetic biology is to create artificial organisms. To achieve this it is essential to understand what life is. Metabolism-replacement systems, or (*M*, *R*)-systems, constitute a theory of life developed by Robert Rosen, characterized in the statement that *organisms are closed to efficient causation*, which means that they must themselves produce all the catalysts they need. This theory overlaps in part with other current theories, including autopoiesis, the chemoton, and autocatalytic sets, all of them invoking some idea of *closure*. A simple model of an (*M*, *R*)-system has been implemented in the computer, and behaves in ways that may shed light on the requirements for a prebiotic self-organizing system. In addition to a trivial steady state in which nothing happens, it can establish a non-trivial steady state in which all intermediates have finite concentrations, with their rates of degradation balanced by their rates of synthesis. The system can be regenerated from the set of food components plus a single intermediate, and maintain itself in that state indefinitely, despite continuous degradation. At the very low compartment volumes that may have existed in prebiotic conditions, for example in cavities in minerals, or in micelles formed by simple amphiphiles, statistical fluctuations in the numbers of molecules need to be taken into account. With the stochastic approach there is no non-trivial steady state in strict mathematical terms, because the system will always collapse to the trivial state after sufficient time. However, the average time before collapse is so long for volumes greater than \(10^{-19}\;{\textsc{l}}\) (much smaller than the volume of the order of \(10^{-15}\;{\textsc{l}}\) for a typical bacterial cell) that for practical purposes the self-maintaining state of non-null concentrations becomes significant, recalling the situation of bistability that is observed in deterministic analysis. In turn, there exists a minimum size below which the self-organizing system cannot maintain itself on chemically relevant time scales. The value of the critical volume depends on the particular concentrations and rate constants assumed, but the principle could apply generally.

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## Notes

- 1.
This sort of approximation was useful when computer time and storage were orders of magnitude more expensive than they are now. Today one would simply store the entire table, or calculate the appropriate value to any desired precision with the proper statistical theory.

- 2.
It is an everyday observation that the chemical output of any organism is different from the food taken in, and it should be obvious from thermodynamic considerations that that must be the case.

- 3.
The collapse would not be totally irreversible if we allow uncatalyzed production of minimal amounts of intermediates: in this case the system could regain the non-trivial steady state if all of the degradation rate constants were sufficiently low.

- 4.
For example, for

*k*= 0.1 s^{−1}, a starting concentration of STU of at least \(0.135\; \text{m}{\textsc{m}}\) would allow the whole system to construct itself. - 5.
This is much smaller than the volume of more than \(10^{-15}\; {\textsc{l}}\; (1\; \upmu\text{m}^3\)) for a bacterium such as

*Escherichia coli*. - 6.
In their recent paper on osmotic structures resembling Leduc’s osmotic gardens, Barge et al. (2012) do not give estimates of the internal volumes of their structures. They have informed us (Barge, personal communication, 8 March 2012) that although these volumes are very difficult to estimate with any accuracy they are certainly orders of magnitude larger than the volumes we are considering here.

- 7.
It is easy to forget that supposedly improbable events occur very frequently. In a human population of 10

^{9}individuals the probability that one of them will experience an event of probability 10^{−9}at any particular moment is close to certainty. - 8.
This is just one illustration of the point that all current theories of life lack some features that an ideal theory ought to have (Letelier et al. 2011). On the other hand the chemical nature of the catalysts—absolutely necessary for the specificity that any self-organizing set of reactions must have—is too vague in autopoiesis and the chemoton to be satisfactory.

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## Acknowledgements

This work was supported by grant BFU2009-12895-C02-02 (Ministerio de Ciencia e Innovación, Spain). GP acknowledges support from PhD scholarship FPU (Formación del Profesorado Universitario, Ministry of Education, Spain). AC-B and MLC were supported by the Centre National de la Recherche Scientifique.

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Cornish-Bowden, A., Piedrafita, G., Morán, F. *et al.* Simulating a Model of Metabolic Closure.
*Biol Theory* **8, **383–390 (2013). https://doi.org/10.1007/s13752-013-0132-0

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### Keywords

- Autopoiesis
- Chemoton
- Hypercycles
- Metabolic closure
- (
*M*,*R*)-systems - Self-organization
- Simulation