Chemical concret machine, detailed (III)

This is a first post about what the chemical concrete machine can do. I concentrate here on geometrical like actions.

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1. Lists and locks.  Suppose you have a family of molecules which you want to free them in the medium in a given order. This corresponds to having a list of molecules, which is “read” sequentially. I shall model this with the help of the zipper from graphic lambda calculus.

Suppose that the molecules we want to manipulate have the form $A \rightarrow A'$, with $A$ and $A'$ from the family of “other molecules” and $\rightarrow$ an arrow, in the model described in Chemical concrete machine, detailed (I).  Here are three zippers (lists).

The first zipper, called a $\beta$ zipper, behaves in the following way. In the presence of $\beta^{+}$ enzymes, there is only one reaction site available, namely the one involving the red and green nodes in the neighbourhood of the $D, D'$. So there is only one reaction possible with a $\beta^{+}$ enzyme, which has a a result the molecule $D \rightarrow D'$ and a new, shorter $\beta$ zipper. This new zipper has only one reaction site, this time involving nodes in the neighbourhood of $C, C'$, so the reaction with the enzyme $\beta^{+}$ gives $C \rightarrow C'$ and a new, shorter zipper. The reaction continues like this, freeing in order the molecules $B\rightarrow B'$, then $A \rightarrow A'$ and $E \rightarrow E'$.

The second zipper is called a FAN-IN zipper (or a $\phi$ zipper). It behaves the same as the previous one, but this time in the presence of the FAN-IN enzyme $\phi^{+}$.

On the third row we see a mixed  zipper. The first molecule $D \rightarrow D'$ is released only in the presence of a $\phi^{+}$ enzyme$, then we are left with a $\beta$ zipper. This can be used to lock zippers. Look for example at the following molecule: called a locked $\beta$ zipper. In the presence of only $\beta^{+}$ enzymes, nothing happens. If we add into the reactor also $\phi^{+}$ enzymes, then the zipper unlocks, by releasing a loop (that’s seen as garbage) and a $\beta$ zipper which starts to react with $\beta^{+}$ enzymes. The same idea can be used for keeping a molecule inactive unless both $\phi^{+}$ and $\beta^{+}$ enzymes are present in the reactor. Say that w have a molecule $A \rightarrow A'$ which is made inactive under the form presented in the following figure The molecule is locked, but it has two reaction sites, one sensible to $\beta^{+}$, the other sensible to $\phi^{+}$. Both enzymes are needed for unlocking the molecule, but there is no preferred order of reaction with the enzymes (in particular these reactions can happen in parallel). _____________________ 2. Sets. Suppose now that we don’t want to release the molecules in a given order. We need to prepare a molecule which has several reaction sites available, so that multiple reactions can happen in parallel, as in the last example. Mathematically, that could be seen as a representation of the set of molecules we want to free, instead of the list of them. This is easy, as described in the next figure: On the first row we see what is called a $\beta$ set. It has 4 possible reaction sites with the enzyme $\beta^{+}$, therefore, in the presence of this enzyme, the molecules $A \rightarrow A'$, … ,$E \rightarrow E’$are released at the same moment. Alternatively, we may think about a $\beta$ set as a bag of molecules which releases (according to the probability of the reaction with a $\beta^{+}$ enzyme$) one of the four molecules $A \rightarrow A'$, … , $D \rightarrow D'$, at random. (It should be interesting to study the evolution of this reaction, because now there are only 3 reaction sites left, …)

On the second row we see a FAN-IN, or $\phi$ set. It behaves the same as the previous one, but this time in the presence of the FAN-IN $\phi^{+}$ enzyme.

Finally, we see a mixed set on the third row. (It should have an interesting dynamics, as a function of the concentrations of the two enzymes.)

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3. Pairs.  Actually, there is no limit but the imagination to what geometrical operations to consider, see for example the posts Sets, lists and order of moves in graphic lambda calculus and Pair of synapses, one controlling the other (B-type NN part III) . As another example, here is a more involved molecule, which produces different pairs of molecules, according to the presence of $\phi^{+}$ or $\beta^{+}$ enzymes.

In the following figure we see how we model a pair of molecules, then two possible reactions a re presented.

The idea is that we can decide, by controlling the amount of $\beta^{+}$ or $\phi^{+}$, to couple $A$ with $D$ and $C$ with $D$, or to couple $A$ with $B$ and $C$ with $D$. Why? Suppose that $A$ and $B$ can react with both $C$ and $D$, depending of course on how close available molecules are.

For example, we want that $A$ and $B$ molecules to be, statistically, one in the proximity of another. We add to the reactor some enzyme $\phi^{+}$ and we obtain lots of pairs $(A,B)_{\beta}$. Now, when we add further the $\beta^{+}$ enzyme, then, immediately after the reaction with this enzyme we are going to have lots of pairs of molecules $A$ and $B$ which are physically close one to another, starting to react.

Instead, if we first introduce $\beta^{+}$ enzymes and then $\phi^{+}$ enzymes, then $A$ would react more likely with $D$.

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