Two direct elaborations of the original model of computational autopoiesis have been published more recently.
Breyer et al.  first relaxed the original restrictions on the motion of bound L particles to allow the formation of flexible chains and, indeed, membranes. Allowing multiple, doubly-bonded, L particles to occupy a single lattice site further enhanced membrane motion. Adopting more sophisticated ``bond re-arrangement'' interactions then allows chain fragments formed within the cell to be dynamically integrated into the membrane. These mechanisms together obviate the need for the chain-based bond inhibition mechanisms--since now, even if L particles spontaneously bond in the interior of the cells, the oligomers so formed remain mobile, and through bond rearrangement, can still successfully function to repair membrane ruptures. In this way, the model is reported as successfully supporting cell growth.
The authors separately consider mechanisms for production of further K particles within a cell. As previously noted, this possibility was absent in the original model, but is, of course, a necessary condition for any development toward self-reproduction. However, the full achievement of self-reproduction is not reported.
Rather similar developments were independently reported by McMullin and Groß . Again, bound L particles were allowed to move, facilitating flexible membranes; a somewhat more complex bond re-arrangement interaction was introduced to facilitate cell growth; and the chain-based bond inhibition interaction was discarded. In this case, however, instead of simply permitting spontaneous L bonding within the cell (and then relying on mobility of these fragments to sustain repair), chain initiation (i.e., bonding between two free L particles) was disabled. This meant that the interior of the cell could still sustain a relatively large concentration of highly mobile free L particles. Other refinements included ``smart'' repair of the membrane (where membrane particles were replaced as soon as they started to decay--even before the membrane was actually ruptured) and ``membrane affinity'' of free L particles. The latter meant that the membrane tended to take a bi-layer form; the outer layer particles being bonded, and the inner layer providing a stock of free L particles to effect very efficient repair.
These modifications together permitted the establishment of autopoietic cells with much greater dynamic stability (longer lifetimes) and/or cells capable of sustained growth. However, it should be noted that, with chain initiation disabled, this particular model would not be capable of exhibiting spontaneous cell generation de novo. Again, this paper identifies the enriched phenomenology as potential steps toward the realisation of self-reproduction of autopoietic cells, through growth and fission; but does not report the achievement of that.
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