# Wnt Signaling Pathway

In a previous post we discussed the significance of chemical reaction networks, the equations that arise from such networks, and our goals in solving them.  Since then, the group has been working on creating a Macaulay2 package that takes a chemical reaction network as input and through various commands gives output reflecting the steady-state equations, a basis for the stoichiometric subspace, etc.  We have created building blocks within the package which correspond to the motifs described in this paper.  The idea is that these motifs can be used to create new reaction networks without the need to input every single reaction.

As an additional example in the package we added the shuttle model of the Wnt signaling pathway described in this paper.  These are the chemical reactions in the model:

Trying to input these reactions (which are not based on any of the motifs we have already created) in our package we ran into a problem when trying to understand how to deal with the empty set present in 4 of the reactions above.

From an algebraic point of view, the two reactions of the form

$x_{**} \xrightarrow{k} x_{**}+\emptyset$

can be viewed without the empty set.  Its significance in the reaction network is that there is  degradation of the protein $\beta$-catenin; however, deleting the $"+ \emptyset"$ part will not change anything in the equations or the properties of the variety.

For reactions of the form

$X \xrightarrow{k} \emptyset~~~(1)$    and    $\emptyset \xrightarrow{k} Y~~(2)$

we have to be more careful.  We think of each complex in the reaction network as a monomial represented by an exponent vector.  In the case above we have 19 species, so each exponent vector will be of dimension $19\times 1$.  For example, the complex $x_2+x_4$ will have ones in the second and fourth position and all other entries will be zero.  But how do we deal with the empty set?  We can think of it as the monomial 1, with exponent vectors of all zeros.  This is significant for reactions of the form $(1)$, since the rate constant $k$ will appear in the steady-state equation for $\dot{X}$.  Aside from this, the empty set does not participate in the stoichiometric or steady-state equations.

From a biology standpoint, the reaction of type $(1)$ represents the degradation of the protein $\beta$-catenin, and the reaction of type $(2)$  – the production of $\beta$-catenin.

From a technical point of view, we need to revise our package so that it accepts an empty set symbol and associates an exponent vector of zeros with it, however, it does not add it to the species list.