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Multiple Bins (2)

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Introduction

We are interested in more biologically realistic situations in which chemical reactions occur "everywhere" along a spatial domain, as opposed to simply at one position. In other words, we have identical reactions occurring at multiple points along some space, with reaction components diffusing along this domain. We are particularly interested in the construction of a stoichiometric-diffusion matrix for such a situation, as well as the characteristics of its fundamental subspaces. As an example, we will consider our open Michaelis-Menten example in the context of bins.


Two Michaelis-Menten Reaction Bins

In order to develop an understanding of the structure of a stoichiometric-diffusion matrix in the case of multiple bins, we begin by examining the simple case of two Michaelis-Menten reaction bins.

We assume that the substrate diffuses into the system from point -1, and then diffuses freely along the spatial domain, which we denote by points 0 and +1. The enzyme and the complex are assumed to remain internal to the reaction area, but freely diffuse along points 0 and +1. The product is then obviously produced at both points 0 and +1, and we will assume that it diffuses to the point +2. Since our diffusion approximation is "pointwise", we must differentiate between points 0 and +1 for substrate, enzyme, complex, and product. Then we have the following "participants" in the reaction: S_{{-1}},S_{0},S_{1},E_{0},E_{1},C_{0},C_{1},P_{0},P_{1},P_{2}. Note also that we assume the reaction and diffusion coefficients to be the same regardless of location.

Chemically, we represent these identical reactions as:


S_{{-1}}{\xrightarrow  {b_{1}}}S_{{0}}+E_{{0}}{\xleftarrow  {v_{{-1}}}}{\xrightarrow  {v_{1}}}C_{{0}}{\xrightarrow  {v_{2}}}E_{{0}}+P_{{0}}{\xrightarrow  {v_{3}}}P_{{1}}
S_{{0}}{\xrightarrow  {b_{1}}}S_{{1}}+E_{{1}}{\xleftarrow  {v_{{-1}}}}{\xrightarrow  {v_{1}}}C_{{1}}{\xrightarrow  {v_{2}}}E_{{1}}+P_{{1}}{\xrightarrow  {v_{3}}}P_{{2}}

<br.>

Using our same diffusion approximation as a gradient, we obtain the following system of differential equations:


{\frac  {\partial [S,-1]}{\partial t}}={\frac  {D_{{S}}}{h^{{2}}}}([S,0]-[S,-1])
{\frac  {\partial [S,0]}{\partial t}}=k_{{-1}}[C,0]-k_{1}[S,0][E,0]+{\frac  {D_{{S}}}{h^{{2}}}}([S,-1]-[S,0])+{\frac  {D_{{S}}}{h^{{2}}}}([S,1]-[S,0])
{\frac  {\partial [S,1]}{\partial t}}=k_{{-1}}[C,1]-k_{1}[S,1][E,1]+{\frac  {D_{{S}}}{h^{{2}}}}([S,0]-[S,1])
{\frac  {\partial [E,0]}{\partial t}}=(k_{{-1}}+k_{2})[C,0]-k_{1}[S,0][E,0]+{\frac  {D_{{E}}}{h^{{2}}}}([E,1]-[E,0])
{\frac  {\partial [E,1]}{\partial t}}=(k_{{-1}}+k_{2})[C,1]-k_{1}[S,1][E,1]+{\frac  {D_{{E}}}{h^{{2}}}}([E,0]-[E,1])
{\frac  {\partial [C,0]}{\partial t}}=k_{1}[S,0][E,0]-(k_{{-1}}+k_{2})[C,0]+{\frac  {D_{{C}}}{h^{{2}}}}([C,1]-[C,0])
{\frac  {\partial [C,1]}{\partial t}}=k_{1}[S,1][E,1]-(k_{{-1}}+k_{2})[C,1]+{\frac  {D_{{C}}}{h^{{2}}}}([C,0]-[C,1])
{\frac  {\partial [P,0]}{\partial t}}=k_{2}[C,0]+{\frac  {D_{{P}}}{h^{{2}}}}([P_{1}]-[P_{0}])
{\frac  {\partial [P,1]}{\partial t}}=k_{2}[C,1]+{\frac  {D_{{P}}}{h^{{2}}}}([P_{0}]-[P_{1}])+{\frac  {D_{{P}}}{h^{{2}}}}([P_{2}]-[P_{1}])
{\frac  {\partial [P,2]}{\partial t}}={\frac  {D_{{P}}}{h^{{2}}}}([P_{1}]-[P_{2}])


Using the same diffusion approximation as mentioned previously, we have generated a size 10\times 12 matrix S:D such that


{\frac  {\partial \phi }{\partial t}}=(S:D){\textbf  {v}},


where


S:D=\left({\begin{smallmatrix}0&0&0&0&0&0&-1&0&0&0&0&0\\1&-1&0&0&0&0&1&-1&0&0&0&0\\0&0&1&-1&0&0&0&1&0&0&0&0\\1&-1&0&0&1&0&0&0&-1&0&0&0\\0&0&1&-1&0&1&0&0&1&0&0&0\\-1&1&0&0&-1&0&0&0&0&-1&0&0\\0&0&-1&1&0&-1&0&0&0&1&0&0\\0&0&0&0&1&0&0&0&0&0&-1&0\\0&0&0&0&0&1&0&0&0&0&1&-1\\0&0&0&0&0&0&0&0&0&0&0&1\end{smallmatrix}}\right)


\phi =(S_{{-1}},S_{0},S_{1},E_{0},E_{1},C_{0},C_{1},P_{0},P_{1},P_{2})


{\textbf  {v}}=(k_{{-1}}[C,0],k_{1}[S,0][E,0],k_{{-1}}[C,1],k_{1}[S,1][E,1],k_{2}[C,0],k_{2}[C,1]
{\frac  {D_{{S}}}{h^{{2}}}}([S,-1]-[S,0]),{\frac  {D_{{S}}}{h^{{2}}}}([S,0]-[S,1]),{\frac  {D_{{E}}}{h^{{2}}}}([E,0]-[E,1]),{\frac  {D_{{C}}}{h^{{2}}}}([C,0]-[C,1])
{\frac  {D_{{P}}}{h^{{2}}}}([P_{0}]-[P_{1}]),{\frac  {D_{{P}}}{h^{{2}}}}([P_{1}]-[P_{2}])



Considering the Reaction Space Only


As we have done previously, note that this method takes into account steady-states for the substrate at point -1 and for the product at point +2, which we are not necessarily concerned with. We will see what happens when these two differential equations are removed from our stoichiometric-diffusion matrix (generating a 8\times 12 matrix):


S:D=\left({\begin{smallmatrix}1&-1&0&0&0&0&1&-1&0&0&0&0\\0&0&1&-1&0&0&0&1&0&0&0&0\\1&-1&0&0&1&0&0&0&-1&0&0&0\\0&0&1&-1&0&1&0&0&1&0&0&0\\-1&1&0&0&-1&0&0&0&0&-1&0&0\\0&0&-1&1&0&-1&0&0&0&1&0&0\\0&0&0&0&1&0&0&0&0&0&-1&0\\0&0&0&0&0&1&0&0&0&0&1&-1\end{smallmatrix}}\right)


\phi =(S_{0},S_{1},E_{0},E_{1},C_{0},C_{1},P_{0},P_{1})


{\textbf  {v}}=(k_{{-1}}[C,0],k_{1}[S,0][E,0],k_{{-1}}[C,1],k_{1}[S,1][E,1],k_{2}[C,0],k_{2}[C,1]
{\frac  {D_{{S}}}{h^{{2}}}}([S,-1]-[S,0]),{\frac  {D_{{S}}}{h^{{2}}}}([S,0]-[S,1]),{\frac  {D_{{E}}}{h^{{2}}}}([E,0]-[E,1]),{\frac  {D_{{C}}}{h^{{2}}}}([C,0]-[C,1])
{\frac  {D_{{P}}}{h^{{2}}}}([P_{0}]-[P_{1}]),{\frac  {D_{{P}}}{h^{{2}}}}([P_{1}]-[P_{2}])


The rank of this matrix is r=7, so the dimension of the right null space is 12-7=5, given by the following orthonormal basis:

V_{{8}}=(0.1148,0.6131,-0.1032,-0.2549,0.2118,0.1349,0.3466,-0.1516,-0.2865,
0.2865,0.2118,0.3466)
V_{{9}}=(0.2957,0.2079,-0.2456,-0.3860,0.2359,-0.4641,-0.2282,-0.1404,0.3237,
-0.3237,0.2359,-0.2282)
V_{{10}}=(0.6182,0.2975,-0.0971,0.0024,-0.4200,0.1987,-0.2213,0.0995,-0.0992,
0.0992,-0.4200,-0.2213)
V_{{11}}=(-0.1210,-0.2837,-0.7147,-0.4051,-0.1035,0.2503,0.1468,0.3096,0.0593,
-0.0593,-0.1035,0.1468)
V_{{12}}=(0.3293,0.1337,-0.0032,0.4616,0.1508,0.1185,0.2693,0.4649,0.3464,
-0.3464,0.1508,0.2693)


Biological Interpretation


We wish to extract biologically useful results from our five-dimensional spanning space. First, note that in general, our first six fluxes -- that is, the reaction fluxes -- must run in the same direction relative to each other to be biologically realistic. For example, in the flux V_{8}, it simply does not make sense for the k_{{-1}}[C,1] and k_{1}[S,1][E,1] reactions to be occurring in "reverse" relative to the rest of the reactions, as this would imply that, for instance, substrate at point 1 is being lost at a rate proportional to the amount of complex at point 1, or that complex at point 1 is being lost at a rate proportional to the amount of substrate and enzyme at point 1, neither of which makes biological sense. Diffusion, on the other hand, can occur in either direction.

Thus, we are looking for a spanning vector that is strictly positive for the first six entries (or strictly negative, as we can simply take the negation of that vector). As we can see, none of these linearly independent spanning vectors by themselves satisfies this requirement. However, we can take a linear combination of any of these vectors to try and generate this situation.

Somewhat arbitrarily, we try V_{8}+-V_{{11}}+V_{{12}}, which produces

V_{8}+-V_{{11}}+V_{{12}}=(0.5651,1.0305,0.8147,0.6118,0.4661,0.0800,0.4691,0.0037,0.0006,
-0.0006,0.4661,0.4691).

Combined with the diffusive fluxes, this situation actually looks fairly realistic. First, note the following flux balances:

{\frac  {D_{{E}}}{h^{{2}}}}([E,0]-[E,1])=-{\frac  {D_{{C}}}{h^{{2}}}}([C,0]-[C,1])
k_{2}[C,0]={\frac  {D_{{P}}}{h^{{2}}}}([P_{0}]-[P_{1}])
{\frac  {D_{{S}}}{h^{{2}}}}([S,-1]-[S,0])={\frac  {D_{{P}}}{h^{{2}}}}([P_{1}]-[P_{2}])

That last flux balance is particularly significant: product is leaving the system at the same rate that the substrate is entering the system. The second flux balance demonstrates that the complex formation into product at point 0 is balanced by the rate at which the product diffuses from point 0 to point 1. The first flux balance is at first somewhat confusing, but makes sense given the known conservation relationship between enzyme and complex in the closed Michaelis-Menten system: namely, there must be constant E+C at both points 0 and 1 in the domain.


n Michaelis-Menten Reaction Bins

We are interested in seeing if this flux balance behavior holds for any number of bins along the spatial domain, approaching the continuum limit where there are an infinite number of bins. Given {\frac  {\partial \phi }{\partial t}}=(S:D){\textbf  {v}} and n points to be considered within the spatial domain, it is clear that the form of the matrix that does not consider the "endpoints" (which do not permit a steady-state flux solution) has:

  • \phi with dimension 4n,
    • each species S, E, C, P considered at each point along the spatial domain
  • {\textbf  {v}} with dimension 3n+2n+2(n-1)=7n-2
    • each point experiencing three essential stoichiometric fluxes k_{1},k_{{-1}}, and k_{2},
    • and diffusive flux gradients for substrate and product between every point (i.e., n), including the endpoints,
    • and diffusive flux gradients for enzyme and complex between only the points of the spatial domain (i.e., n-1)


MATLAB Code
We can write a (rather inefficient) script to generate a matrix for us for any n:


We will organize our stoichiometric-diffusion matrix similarly to our n=2 case above:

  • \phi =(S_{0},...,S_{{n-1}},E_{0},...,E_{{n-1}},C_{0},...,C_{{n-1}},P_{0},...,P_{{n-1}})
    • again, S_{{-1}} and P_{{n}} represent our endpoints and their fluxes are not considered for the purposes of our flux balance analysis
  • {\textbf  {v}}=(k_{{-1}}[C,0],...,k_{{-1}}[C,(n-1)],k_{1}[S,0][E,0],...,k_{1}[S,(n-1)][E,(n-1)],k_{2}[C,0],...,k_{2}[C,(n-1)],
{\frac  {D_{{S}}}{h^{{2}}}}([S,-1]-[S,0]),{\frac  {D_{{S}}}{h^{{2}}}}([S,0]-[S,1]),{\frac  {D_{{E}}}{h^{{2}}}}([E,0]-[E,1]),{\frac  {D_{{C}}}{h^{{2}}}}([C,0]-[C,1])
{\frac  {D_{{P}}}{h^{{2}}}}([P_{0}]-[P_{1}]),...,{\frac  {D_{{S}}}{h^{{2}}}}([S,(n-2)]-[S,(n-1)]),{\frac  {D_{{E}}}{h^{{2}}}}([E,(n-2)]-[E,(n-1]),
{\frac  {D_{{C}}}{h^{{2}}}}([C,(n-2)]-[C,(n-1)]),{\frac  {D_{{P}}}{h^{{2}}}}([P_{(}n-2)]-[P_{(}n-1)]),{\frac  {D_{{P}}}{h^{{2}}}}([P_{1}]-[P_{2}]))


% Creates a Michaelis-Menten stoichiometric-diffusion matrix for a given number of desired approximation points n.

%{
File: create_mm_sd.m
Author: Sam Hsu

INPUT: 
  n: number of approximation points
OUTPUT:
  mm_SD: Michaelis-Menten stoichiometric-diffusion matrix
%}


function[mm_SD] = create_mm_sd(n)

% Dimension of matrix must be 4n x 7n-2
mm_SD = zeros(4*n,(7*n)-2);

% Stoichiometric Matrix
% for k_{-1} reaction columns
for a = 1:1:n
    mm_SD(a,a) = 1;
    mm_SD(a+2,a) = 1;
    mm_SD(a+4,a) = -1;
end

% for k_{1} reaction columns
for b = 1:1:n
    mm_SD(b,n+b) = -1;
    mm_SD(b+2,n+b) = -1;
    mm_SD(b+4,n+b) = 1;
end

% for k_{2} reaction columns
for c = 1:1:n
    mm_SD(n+c,2*n+c) = 1;
    mm_SD(n+c+2,2*n+c) = -1;
    mm_SD(n+c+4,2*n+c) = 1;
end

% Diffusion Matrix
% for substrate diffusion column 0
mm_SD(1,3*n+1) = 1;

% for substrate diffusion columns 1,...,n-1
for d = 1:1:(n-1)
    mm_SD(d,3*n+1+d) = -1;
    mm_SD(d+1,3*n+1+d) = 1;
end

% for enzyme diffusion columns 1,...,n-1
for e = 1:1:(n-1)
    mm_SD(n+e,3*n+(n-1)+1+e) = -1;
    mm_SD(n+e+1,3*n+(n-1)+1+e) = 1;
end

% for complex diffusion columns 1,...,n-1
for f = 1:1:(n-1)
    mm_SD(2*n+f,3*n+2*(n-1)+1+f) = -1;
    mm_SD(2*n+f+1,3*n+2*(n-1)+1+f) = 1;
end

% for product diffusion columns 1,...,n-1
for g = 1:1:(n-1)
    mm_SD(3*n+g,3*n+3*(n-1)+1+g) = -1;
    mm_SD(3*n+g+1,3*n+3*(n-1)+1+g) = 1;
end

% for product diffusion column n
mm_SD(4*n,7*n-2) = -1;


Structure of Relevant Spanning Space Vectors
Equipped with this, the goal is then to find the general flux balance structure of biologically relevant spanning space vectors for a given matrix with n approximation points. We have already observed that the stoichiometric fluxes should be strictly positive.