# Metabolism of loops (a chemical reaction network in the chemical concrete machine)

In the following figure you see a chemical reaction network in the chemical concrete machine which involves only “+” enzymes.

• the molecule from the left upper corner corresponds to combinator $\Omega = (\lambda x . (xx)) (\lambda x . (xx))$
• for each molecule I marked the available reaction sites, with dashed red closed curves, along with the name in red of the enzyme which is interested in the respective reaction site,
• for the chemical reactions, figured with blue arrows, I put the name of the move which corresponds to the respective chemical reaction, with an added “+” to indicate that the reaction is unidirectional, as if only “+” enzymes are available (that is I wrote “$DIST^{+}$” instead of “$\delta^{+}$” and so on, see for details the notations used in the chemical concrete machine),
• whenever there are non-overlapping reaction sites, we can perform in parallel the moves (reactions),
• but in some places we have overlapping reaction sites, like in the case of the molecule from the 3rd row, left, where a $\beta$ reaction site and a $DIST$ reaction site overlap.
• In case of overlapping reaction sites there are multiple possibilities, which produce branches in the chemical reaction network,
• I have not used any elimination of loops,
• several molecules from the figure don’t correspond to lambda calculus terms, thus what  is figured is not a representation of the usual fact that the combinator $\Omega$ does not have a normal form (one which cannot be reduced further),
• in the middle of the figure we see a loop-producing cycle, hence the name,
• curiously, by going outside lambda calculus, we can reduce $\Omega$ (at left upper corner) to a “normal” form (right lower corner), i.e. a loop and a molecule which does not have any “+” reaction sites. (The green fork node of that molecule is a fan-out node and the molecule is like a fan-out gate with one output curling back to the input of the fan-out, but remember that no signals circulate through the arrows 🙂  )

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UPDATE:  For clarity, here is the usual cycle of reductions of the combinator $\Omega$, seen in graphic lambda calculus (but with the drawing conventions of the chemical concrete machine):

(A similar figure appears after the proof of Theorem 3.1  arXiv:1305.5786v2 [cs.LO] . )

In graphic lambda calculus we have the GLOBAL FAN-OUT move, which, as the name says, is a global move. In the chemical concrete machine we have only local moves.  Here is how the same cycle is achieved in the chemical concrete machine.

You see a move (reaction) which is labelled “(multiple) CO-ASSOC”. The reason is that we need several CO-ASSOC moves to pass from the molecule at the right of the 2nd row to the one from the 3rd row. Alternatively, the same can be done with a SWITCH move, which is a succession of FAN-IN(+) , CO-COMM(+)  and FAN-IN(-), therefore unpleasant as well. Moreover, if we invent a SWITCH enzyme (i.e. if we accept SWITCH as a new move which does not modify the whole chemical machine, because SWITCH is a consequence of the other moves) then we have an explosion of places where SWITCH enzyme could act.

In conclusion, the usual endless reduction of $\Omega$ in lambda calculus, is possible, but highly unlikely in the chemical concrete machine, and only in the presence of CO-ASSOC or SWITCH enzymes, and moreover under tight control of the places where these enzymes act.

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