Tag Archives: synthetic biology

Neurons know nothing, however …

… they know surprisingly much, according to the choice of definition of “neural knowledge”. The concrete definition which I adopt is the following:  the knowledge of a neuron at a given moment is the collection (multiset) of molecules it contains. The knowledge of a synapse is the collection of molecules present in respective axon, dendrite and synaptic cleft.

I take the following hypotheses for a wet neural network:

  • the neural network is physically described as a graph with nodes being the neurons and arrows being the axon-dendrite synapses. The network is built from two ingredients: neurons and synapses. Each synapse involves three parts: an axon (associated to a neuron), a synaptic cleft (associated to a local environment) and a dendrite (associated to a neuron).
  • Each of the two ingredients of a neural network, i.e. neurons and synapses, as described previously, function by associated chemical reaction networks, involving the knowledge of the respective ingredients.
  • (the most simplifying hypothesis)  all molecules from the knowledge of a neuron, or of a synapse, are of two kinds: elements of MOLECULES or enzyme  names from the chemical concrete machine.

The last hypothesis seem to introduce knowledge with a more semantic flavour by the backdoor. That is because, as explained in  arXiv:1309.6914 , some molecules (i.e. trivalent graphs from the chemical concrete machine formalism) represent lambda calculus terms. So, terms are programs, moreover the chemical concrete machine is Turing universal, therefore we end up with a rather big chunk of semantic knowledge in a neuron’s lap. I intend to show you this is not the case, in fact a neuron, or a synapse does not have (or need to) this kind of knowledge.

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Before giving this explanation, I shall explain in just a bit more detail how the wet neural network,  which satisfies those hypotheses, works.  A physical neuron’s behaviour is ruled by the chemistry of it’s metabolic pathways. By the third hypothesis these metabolic pathways can be seen as graph rewrites of the molecules (more about this later). As an effect of it’s metabolism, the neuron has an electrical activity which in turn alters the behaviour of the other ingredient, the synapse. In the synapse act other chemical reaction networks, which are amenable, again by the third hypothesis, to computations with the chemical concrete machine. As an effect of the action of these metabolic pathways, a neuron communicates with another neuron. In the process the knowledge of each neuron (i.e. the collection of molecules) is modified, and the same is true about a synapse.

As concerns chemical reactions between molecules, in the chemical concrete machine formalism there is only one type of reactions which are admissible, namely the reaction between a molecule and an enzyme. Recall that if (some of the) molecules are like lambda calculus terms, then (some of the) enzymes are like names of reduction steps and the chemical reaction between a molecule and an enzyme is assimilated to the respective reduction step applied to the respective lambda calculus term.

But, in the post  SPLICE and SWITCH for tangle diagrams in the chemical concrete machine    I proved that in the chemical concrete machine formalism there is a local move, called SWITCH

chem_switch

which is the result of 3 chemical reactions with enzymes, as follows:

chem_switch_2

Therefore, the chemical concrete machine formalism with the SWITCH move added is equivalent with the original formalism. So, we can safely add the SWITCH move to the formalism and use it for defining chemical reactions between molecules (maybe also by adding an enzyme, or more, for the SWITCH move, let’s call them \sigma).  This mechanism gives chemical reactions between molecules with the form

A + B + \sigma \rightarrow C + D + GARB

where $\latex A$ and B are molecules such that by taking an arrow from A and another arrow from B we may apply the \sigma enzyme and produce the SWITCH move for this pair of arrows, which results in new molecules C and D (and possibly some GARBAGE, such as loops).

In conclusion, for this part concerning possible chemical reactions between molecules, we have enough raw material for constructing any chemical reaction network we like. Let me pass to the semantic knowledge part.

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Semantic knowledge of molecules. This is related to evaluation and it is maybe the least understood part of the chemical concrete machine. As a background, see the post  Example: decorations of S,K,I combinators in simply typed graphic lambda calculus , where it is explained the same phenomenon (without any relation with chemical metaphors) for the parent of the chemical concrete machine, the graphic lambda calculus.

Let us consider the following rules of decorations with names and types:

chem_decor_1

If we consider decorations of combinator molecules, then we obtain the right type and identification of the corresponding combinator, like in the following example.

chem_decor_2

For combinator molecules, the “semantic knowledge”, i.e. the identification of the lambda calculus term from the associated molecule, is possible.

In general, though, this is not possible. Consider for example a 2-zipper molecule.

chem_decor_3

We obtain the decoration F as a nested expression of A, D, E, which enough for performing two beta reductions, without knowing what A, D, E mean (without the need to evaluate A, D, E). This is equivalent with the property of zippers, to allow only a certain sequence of graphic beta moves (in this case two such moves).

Here is the tricky part: if we look at the term F then all that we can write after beta reductions is only formal, i.e. F  reduces to (A[y:=D])[x:=E], with all the possible problems related to variable names clashes and order of substitutions. We can write this reduction but we don’t get anything from it, it still needs further info about relations between the variables x, y and the terms A, D, E.

However, the graphic beta reductions can be done without any further complication, because they don’t involve any names, nor of variables, like x, y, neither of terms, like A, D, E, F.

Remark that the decoration is made such that:

  • the type decorations of arrows are left unchanged after any move
  • the terms or variables decorations (names elsewhere “places”) change globally.

We indicate this global change like in the following figure, which is the result of the sequence of the two possible \beta^{+} moves.

chem_decor_4

Therefore, after the first graphic beta reduction, we write  A'= A[y:=D] to indicate that A' is the new, globally (i.e. with respect to the whole graph in which the 2-zipper is a part) obtained decoration which replaces the older A, when we replace y by D. After the second  graphic beta reduction we use the same notation.

But such indication are even misleading, if, for example, there is a path made by arrows outside the 2-zipper, which connect the arrow decorated by D with the arrow decorated by y.  We should, in order to have a better notation, replace D by D[y:=D], which gives rise to a notation for a potentially infinite process of modifying D. So, once we use graphs (molecules) which do not correspond to combinators (or to lambda calculus terms), we are in big trouble if we try to reduce the graphic beta move to term rewriting, or to reductions in lambda calculus.

In conclusion for this part: decorations considered here, which add a semantic layer to the pure syntax of graph rewriting, cannot be used as replacement the graphic molecules, nor should reductions be equated with chemical reactions, with the exception of the cases when we have access to the whole molecule and moreover when the whole molecule is one which represents a combinator.

So, in this sense, the syntactic knowledge of the neuron, consisting in the list of it’s molecules, is not enough for deducing the semantic knowledge which is global, coming from the simultaneous decoration of the whole chemical reaction network which is associated to the whole neural network.

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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:

ess_4

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^{-}.

ess_5

(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:

ess_6

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.

convention_2

convention_3

convention_6

convention_4

The last move, elimination of loops,

convention_5

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.