The Y A_k Ψ representation

This converts the stoichiometric representation dx/dt = S v(x) of a chemical
reaction network to the form 
S v(x) = Y A_k Ψ(x) = Y G K Ψ(x).

Here A_k is the Laplacian of the complexes graph and Y is the complexes to species matrix. The factoring A_k = G K is found in 
V. Katsnelson's UCSD undergrad honors thesis.
It also computes properties of A_k and the deficiency of the network.

(* This notebook presents all the commands available in the
chemYAK.m file. *)

<<"Desktop/chemYAk.m"

S=Transpose[{{-1,1,0,0,1,1},{-1,1,0,0,0,0},{-1,0,1,1,0,0},{-1,1,-1,1,0,0},{0,0,-1,1,-1,0},{1,-1,0,0,0,0},{0,0,1,-1,0,0}}](*stoichiometric matrix*); 
S //MatrixForm

(* matrixY takes the Stoichiometric matrix as intput and yields a 
list of three matrices. 
The first is matrix Y, the second is a list of the input complexes for all 
the reactions (with repetitions), and the third is a list of the output 
complexes for all the reactions (with repetitions). *)

Y = matrixY[S];

(* Here is the matrix Y. *)

MatrixForm[Y[[1]]]

(* Here are the input complexes. They may be identified with
monomials whose exponents are given by the rows of this matrix. *)

MatrixForm[Transpose[Y[[2]]]]

(* Here are the output complexes. *)

MatrixForm[Transpose[Y[[3]]]]

(* matrixG takes in matrix Y, the input complexes matrix, 
and the output complexes matrix, and yields matrix G. *)

G = matrixG[Y];
G//MatrixForm

(* Check to make sure YG=S *)

Print["YG = ", MatrixForm[Y[[1]].G], ",  S=" ,MatrixForm[S]];
Y[[1]].G==S

Psi = makeMonomial[-Y[[1]],x];
Print["Ψ(x)= ",MatrixForm[Psi]]

(* matrixK takes matrix G as input and yields matrix K. *)

K = matrixK[G];
K//MatrixForm

Ak=G.K;
Ak//MatrixForm

** We finally obtain our desired formula **

Print["YGKΨ(x)= ",MatrixForm[Y[[1]]],MatrixForm[G],MatrixForm[K],MatrixForm[Psi]]

(* The desired decomposition Y Ak Ψ(x). *)

decomposition[S]

(* linkageClasses takes the Laplacian of a graph (e.g. the A_k matrix in the Y A_k Ψ(x)) as an input. It returns a vector with two components. The first component gives a list of all the linkage classes. For example, {1,3,4} would be a linkage class where the first, third and fourth complex participate in reactions with each other. By definition, the linkage classes are disjoint. 

The second component in our output is a list of 3-tuples for each different complex in our chemical network. A 3-tuple will list all the reactions a certain complex particpates in. The first vector in the 3-tuple gives the index i of a particular complex y_i. The second vector in the 3-tuple gives indices for of each complex that y_i particates in a reaction with, and where y_i is an input for that reaction.
The third vector in the 3-tuple gives indices for of each complex that y_i particates in a reaction with, and where y_i is an output for that reaction. For example, the 3-tuple {{5},{1,2},{2,4,6}} means we have the reactions y_5->y_1, y_5->y_2, y_2->y_5, y_4->y_5, and y_6->y_2. *)

components=linkageClasses[Ak]

Print["The number of linkage classes is ",Length[components[[1]]]]

(*This gives the topological deficiency of a chemical reaction
network. The input is the stoichiometric matrix. *)

topDeficiency[S]

(* The next formula gives the chemical deficiency of a chemical reaction
network. Its input is the stoichiometric matrix. *)

matDeficiency[S]

(* Another example done from scratch: t.brucei. *)

S={{0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{-1,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,-1,0,0},{0,0,0,0,1,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-2,2,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0},{0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,1,-1,1,-1,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,-1,1,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,-1,1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,1,-1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,-2,2},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,-1,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-(1/2),(1/2),0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0},{1,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,0,0}};
{Y,Ak,Psi}=decomposition[S];
Y//MatrixForm
Ak//MatrixForm
Psi//MatrixForm
Print["The topological deficiency is ",topDeficiency[S]];
Print["The matrix deficiency is ",matDeficiency[S]];
Print["The linkage classes: ",linkageClasses[Ak]];

(* A (too) big example *)

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{Y,Ak,Psi}=decomposition[S];
Y//MatrixForm
Ak//MatrixForm
Psi//MatrixForm
Print["The topological deficiency is ",topDeficiency[S]];
Print["The matrix deficiency is ",matDeficiency[S]];
Print["The linkage classes: ",linkageClasses[Ak]];