# Chemical concrete machine, detailed (II)

Continues from the  Chemical concrete machine, detailed (I).  In this part I connect with  the previous post  Chemical concrete machine, short list of gates and moves, then there is a discussion part.

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Let’s look again at the two examples of chemical reactions from the first part:

These reactions look as being complementary, as if one is the inverse of another. Indeed, this is the case, but for seeing this better we need to refine a bit the notations.

For each reaction is specified an enzyme, $\beta^{+}$ for the first reaction and $\beta^{-}$ for the second reaction, and moreover a reaction site, denoted by closed red, dashed, curve. The meaning is that the enzyme react with the molecule by binding at the reaction site and by modifying it, without any other action on the rest of the molecule.

Practically, each enzyme changes a small preferred pattern into another. The rest of the molecule does not take part to the reaction and for this reason is better to concentrate exclusively on the reaction sites and how they change.

The next figure represents this change, for the pair of complementary enzymes $\beta^{+}$ and $\beta^{-}$.

(The crossing from the right hand side of the figure is not real, remember that the molecules float in a 3D container. The 2D drawings we make are projections in the plane of what we see in space.)

The drawing is made such that to make clear the correspondence between  the free ends or beginnings of arrows pointing or coming from the rest of the space.

Now is clear why the two reactions are one the inverse of another. Finally, we represent both reactions like this:

In graphic lambda calculus, this is the most important move, namely the graphic beta move. It corresponds to the beta reduction from lambda calculus.

As simple as it might seem to the eyes of a biochemist, it has tremendous powers. Besides (graphic lambda calculus), it appears also in topology research related to various theories in quantum physics, see the UNZIP move described in   The algebra of knotted trivalent graphs and Turaev’s shadow world  .

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I can now describe the allowed chemical reactions with enzymes in the model of the chemical concrete machine. I reproduce further the last figure from the post Chemical concrete machine, short list of gates and moves , with the understanding that it describes such chemical reactions, with the notational conventions detailed here.

Reactions with enzymes are called “moves”, as in graphic lambda calculus. Moreover, for each reaction-move is given a link to the description of the said move in graphic lambda calculus.

The last move, elimination of loops,

is not a chemical reaction, however! It’s meaning is that loops can be considered garbage (element of GARB). This is in contradiction with the assumption that GARB consists of reaction products which don’t react further with the molecules. Sometimes we might need reactions between loops and enzymes, like $\beta^{-}$.   This is a subject which will be discussed later, but the zest of it is that GARB may also be seen as the collection of reaction products which we don’t want to detect at the end of the chain of reactions.

The list of allowed chemical reactions is minimal. We may add other reactions, like reactions between the “other molecules”, this will also be discussed later.

I the next post I shall describe what the chemical machine can do, in principle.

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Discussion.    We have now at our disposal an universe of molecules and a family of allowed chemical reactions. In order to get a usable model we need a bit more. The natural addition would be to put these reaction in the form of a chemical reactions network, or, equivalently,  a Petri net. This is a path to follow, which will give explicit, numerical, predictions about the behaviour of the chemical concrete machine.

We cannot apply directly Petri nets to the situation at hand, because the chemical reactions, as described here, are between reaction sites, and not between molecules. A molecule may have several reaction sites, moreover there are reaction sites which involve two molecules. After each possible reaction with an enzyme, the resulted molecule, or molecules, may have several reaction sites as well.

Moreover, a reaction site, say, which involves two molecules, like the reaction site containing two arrows, possibly from different molecules, are aplenty. We need to make physical assumptions like saying that a reaction site might be localized in the 3D space, otherwise we have a combinatorial explosion of possible reactions involving one, or a pair of molecules.

These problems related to reaction sites have surely been studied elsewhere. One possible place to learn from might be (in the view of a naive mathematician like me), the research around the  SCHEMA algorithm.   Are reaction sites schemas?

This brings us to the question of the real implementation of the chemical concrete machine. It is clear that the subject belongs to the synthetic biology field and that there is a strong need for a collaborative help from specialists in this field.

There are two possible ways for such an implementation:

• to construct molecules like the essential ones, with the purpose of manipulating interesting other molecules, which appear in the model as the “other molecules”. Indeed, as we shall see, the chemical concrete machine can perform logical computation (it is Turing complete) in a much simpler form than imitating the structure of a silicon computer (instead, it uses extreme functional programming), and also it can perform geometrical operations, like hoarding and grouping interesting “other” molecules, releasing them at the detection of a chemical signal, and so on, again in a much simpler way than by imitating a mechanical large scale machine, or a silicon computer with actuators and effectors.
• or to recognize chemical reactions in the reality, for example in a living organism, which satisfy the chemical concrete machine formalism. In this second case, which might be likely due to the simplicity of the formalism, the information we get from the chemical concrete machine model is semantic, eg. if we want to “convince”  a living cell to do a certain logical computation, we may exploit the chemical concrete machine formalism, embodied in the cell by recognizing the reactions which are meaningful for the formalism, and then trying to find out how to exploit them (basically how to detect or amplify their effects). This second way seems to be related to the extremely interesting path opened by Fontana and Buss , saying that essentially lambda calculus is something a sufficiently complex chemical reaction  network manifests.