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Conceived and designed the experiments: G. Piedrafita, F. Montero, F. Morán, M.L. Cárdenas, A. Cornish-Bowden. Performed the experiments: G. Piedrafita. Analyzed the data: G. Piedrafita, F. Montero, F. Morán, M.L. Cárdenas, A. Cornish-Bowden. Wrote the paper: M.L. Cárdenas, A. Cornish-Bowden.

The authors have declared that no competing interests exist.

A living organism must not only organize itself from within; it must also maintain its organization in the face of changes in its environment and degradation of its components. We show here that a simple (M,R)-system consisting of three interlocking catalytic cycles, with every catalyst produced by the system itself, can both establish a non-trivial steady state and maintain this despite continuous loss of the catalysts by irreversible degradation. As long as at least one catalyst is present at a sufficient concentration in the initial state, the others can be produced and maintained. The system shows bistability, because if the amount of catalyst in the initial state is insufficient to reach the non-trivial steady state the system collapses to a trivial steady state in which all fluxes are zero. It is also robust, because if one catalyst is catastrophically lost when the system is in steady state it can recreate the same state. There are three elementary flux modes, but none of them is an enzyme-maintaining mode, the entire network being necessary to maintain the two catalysts.

The question of whether a whole organism (as opposed to particular properties of an organism) can be modeled in the computer has been controversial. As a step towards resolving it, we have studied the feasibility of simulating the behavior of a simple theoretical model in which all the catalysts needed for the metabolism of a system are themselves products of the metabolism itself, and in which there is a continuous loss of catalysts in unavoidable degradation reactions. In addition to a trivial (“dead”) steady state in which all rates are zero, the model is capable of establishing a stable non-trivial steady state with finite and reproducible fluxes. This can be achieved by “seeding” it with a sufficient quantity of at least one of the catalysts needed for functioning. It is also robust, because it can recover from a catastrophic disappearance of a catalyst.

Several theories of life

To give concrete expression to the idea of an (M,R)-system , and to evaluate its possible relevance to the origin of life, we proposed

(a) The metabolites shown inside squares (input) are considered to be “external” and to have fixed concentrations. The reactions shown in red constitute the metabolic process, those in blue the replacement process, and in gray the replacement of the replacement catalyst. (b) Expanded version of the model in which each catalyzed reaction is expanded into a cycle of three chemical reactions with explicit rate constants. Each forward rate constant refers to the reaction in the direction of the arrow, and the three degradation reactions, steps 4, 8 and 11, are assumed to be uncatalyzed and irreversible. All rate constants are treated as constant with the values shown, apart from

A controversial aspect of Rosen's analysis is his contention that a system closed to efficient causation cannot have computable models

We shall show that a simple (M,R)-system can be robust, capable of a recovering from the loss of most of its catalysts, and in addition has the interesting property of bistability. As Delbrück

For the system to be simulated it needs to be defined in precise numerical terms, and for doing this it is convenient to expand the catalytic processes shown in

All simulations and studies of the stability of the steady states found were done with Matlab and checked with COPASI

As we shall be supposing that the system in

In this context it is important to note that organizational closure does not imply thermodynamic closure, or vice versa. In the Aristotelean terminology favored by Rosen

The concentration evolution of the different metabolites in

Stationary solutions of the system of

For

The model was simulated for

STU is not of course the only catalytic intermediate that could be used for seeding the system, and results with each of the others, and for some pairs of intermediates, are shown in

Seed | ||||

Minimum initial concentration | Minimum initial concentration | |||

STU | 0.135 | 15.63 | 11.460 | 14.32 |

STUS | 0.135 | 15.63 | 11.374 | 14.32 |

STUST | 0.135 | 15.63 | 11.378 | 14.32 |

ST | — | no steady state | — | no steady state |

SU | — | no steady state | — | no steady state |

SUST | 0.355 | 15.63 | 9.896 | 14.32 |

SUSTU | 0.278 | 15.63 | 8.845 | 14.32 |

STUSU | 0.114 | 15.63 | 6.801 | 14.32 |

STU+SU | 0.099 | 15.63 | 5.251 | 14.32 |

ST+SUST | 0.295 | 15.63 | 8.143 | 14.32 |

SU+STUS | 0.099 | 15.63 | 5.184 | 14.32 |

ST+SU | 0.455 | 15.63 | 10.433 | 14.32 |

The reason why ST and SU cannot act as seed can be seen by inspection of

To verify the stability of the steady states, the Jacobian matrices were evaluated at the steady states obtained, and the eigenvectors and eigenvalues calculated. For those conditions in which three steady states were obtained,

For

The diagram of

The unstable steady state that appears in those conditions of bistability,

The calculation refers to

Simulations were done with

It is clear that the system as described is capable of reaching a stable non-trivial steady state with finite fluxes and finite concentrations of all intermediates. However, before it can be regarded as a useful model of a self-maintaining system, and thus relevant to the early stages of metabolic evolution, it needs to be shown to be capable of recovering from catastrophic loss of one or more catalysts. To test this, it was allowed to reach the non-trivial stable steady state characteristic of

The figure shows the time evolution of the system, starting from the stable non-trivial steady state for

As STU catalyzes two different processes (synthesis both of SU and of ST), loss of STU is clearly the most stringent loss of catalyst one could consider, but for completeness we also tested the effect of loss of all forms of SU, with similar results. All of this shows that the system is highly robust, not only for infinitesimal perturbations, as tested by analysis of the Jacobian matrix, but also for large perturbations. Unless it is perturbed to such a large extent that the separating barrier mentioned is crossed, e.g. below the threshold requirements listed in

With the use of MetaTool

(a) The model contains five reaction subsets, consisting of reaction 1 (red), reactions 2 and 3 (magenta), reactions 4, 5, 6 and 7 (blue), reactions 8, 9 and 10 (green); and reaction 11 (gray). (b) There are three elements of the basis, one consisting of reactions 1, 2, 3 and 11 (

The resulting convex basis can be expressed in the following way:

To study the relative contributions of the basis elements to the steady-state flux distribution,

(a)

In the present model, the elementary flux modes coincide with the elements of the convex basis. Nevertheless, none of them is an enzyme-maintaining mode

A simple model of an (M,R)-system consisting of three catalytic cycles organized so that all catalysts are products of reactions within the system is able to establish and maintain a non-trivial steady state capable of resisting degradation of all the catalysts, provided that this degradation is not so fast that the catalysts are eliminated faster than they are regenerated. This model was originally proposed as a way of giving concrete expression to the abstract view of life embodied in Rosen's (M,R)-system s

As our original model

As mentioned in the

The simplicity of this robust self-maintaining system and its capacity to be easily seeded may allow us to regard it as a plausible prebiotic system. Specifically, the establishment of a reflexive autocatalysis, i.e. autocatalysis that results from the structure of the whole network rather than from specifically autocatalytic components, is a typical common feature of models that illustrate recent theories of the origin of life; for example, the “lipid-world” scenario

In this analysis we have effectively assumed that a primitive self-maintaining system has metabolism but does not have information processing, in other words a metabolism-first scenario for the origin of life. All of the principal current theories of life