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Multiple Bins

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Introduction

Biological systems of varying complexities are frequently simplified by division into bins, generally based on spatial domains. These bins are frequently connected by diffusion reactions, as products from one reaction (in one bin) flow to a different spatial domain (a different bin) and are taken up as a reactant in a separate reaction. We are particularly interested in the composition of a stoichiometric-diffusion matrix for multiple bins.

As an example, we consider our open Michaelis-Menten example in the context of bins. Points -1, 0, and +1 can be in a sense taken as representative of three separate bins, where the -1 bin "holds" the substrate, which flows into the 0 bin. The 0 bin "holds" the main Michaelis-Menten reactions, involving the substrate and enzyme combining into a complex and subsequently forming a product. Finally, the +1 bin "holds" the product leaving the reaction domain to fulfill whatever nefarious purposes it might have.

It is worth noting that our method of incorporating diffusion into our stoichiometric matrix using the second derivative and Fick's Laws involves the diffusing substance being "held" at a fixed distance h, as opposed to the biologically realistic case of various distances radially around the fixed enzyme. In that sense, the -1 bin "holding" the substrate and the +1 bin "holding" the product might be better described as existing around the 0 reaction bin, as opposed to existing to the left and to the right, respectively. Additionally, note that the reaction gradient method of approximating diffusion as a flux automatically accounts for "net" diffusion between bins, including in the stoichiometric matrix. For example, the substrate diffusion term {\frac  {D_{{S}}}{h^{{2}}}}([S,-1]-[S,0]) represents diffusion into the reaction stoichiometry when the amount of substrate in the -1 bin is greater, and represents diffusion out of the reaction stoichiometry when the amount of substrate in the 0 bin is greater.


Two Connected Michaelis-Menten Reactions

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 connected Michaelis-Menten reactions, where the product of the first reaction forms the substrate for a second substrate-enzyme reaction. Such a cascade is relevant to our interest in the complement cascade found in the human innate immune system. We can then consider the system as being composed of four distinct bins: the -1 bin again holds the substrate which reacts in the first Michaelis-Menten reaction, which we denote S. The enzyme and the complex for the first reaction are denoted E_{{1}} and C_{{1}}, respectively, and will be assumed to remain native to the 0 bin, which will hold the first Michaelis-Menten reaction. The product produced in this scenario is denoted P_{{1}} and flows into the +1 bin, where the second Michaelis-Menten reaction occurs. The enzyme and the complex for the second reaction are denoted E_{{2}} and C_{{2}}, respectively, and will be assumed to remain native to the +1 bin. Finally, the +2 bin will hold the outflow of the product of the second Michaelis-Menten reaction, denoted P_{{2}}. We keep the same reaction names for the first Michaelis-Menten reaction and assign k_{{-2}},k_{{3}},k_{{4}} as the corresponding reactions for the second Michaelis-Menten reaction.

We formulate our system of differential equations:


{\frac  {\partial [S,0]}{\partial t}}=k_{{-1}}[C_{1}]-k_{1}[S,0][E_{{1}}]+{\frac  {D_{{S}}}{h^{{2}}}}([S,-1]-[S,0])
{\frac  {\partial [S,-1]}{\partial t}}={\frac  {D_{{S}}}{h^{{2}}}}([S,0]-[S,-1])
{\frac  {\partial [E_{{1}}]}{\partial t}}=(k_{{-1}}+k_{2})C_{{1}}-k_{1}[S,0][E_{{1}}]
{\frac  {\partial [C_{{1}}]}{\partial t}}=k_{1}[S,0][E_{{1}}]-(k_{{-1}}+k_{2})[C_{{1}}]
{\frac  {\partial [P_{{1}},0]}{\partial t}}=k_{2}[C_{{1}}]+{\frac  {D_{{P_{{1}}}}}{h^{{2}}}}([P_{{1}},+1]-[P_{{1}},0])
{\frac  {\partial [P_{{1}},+1]}{\partial t}}=k_{{-2}}[C_{2}]-k_{3}[P_{1},+1][E_{2}]+{\frac  {D_{{P_{{1}}}}}{h^{{2}}}}([P_{{1}},0]-[P_{{1}},+1])
{\frac  {\partial [E_{{2}}]}{\partial t}}=(k_{{-2}}+k_{4})C_{{2}}-k_{3}[P_{{1}},+1][E_{{2}}]
{\frac  {\partial [C_{{2}}]}{\partial t}}=k_{3}[P_{{1}},+1][E_{{2}}]-(k_{{-2}}+k_{4})[C_{{2}}]
{\frac  {\partial [P_{{2}},+1]}{\partial t}}=k_{4}[C_{{2}}]+{\frac  {D_{{P_{{2}}}}}{h^{{2}}}}([P_{{2}},+2]-[P_{{2}},+1])
{\frac  {\partial [P_{{2}},+2]}{\partial t}}={\frac  {D_{{P_{{2}}}}}{h^{{2}}}}([P_{{2}},+2]-[P_{{2}},+1])


Using the same diffusion approximation as mentioned previously, we have generated a size 10\times 9 matrix S:D such that


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


where


S:D={\begin{pmatrix}-1&1&1&0&0&0&0&0&0\\1&-1&-1&0&0&0&0&0&0\\-1&1&0&1&0&0&0&0&0\\0&0&0&-1&0&0&0&0&0\\0&0&1&0&-1&0&0&0&0\\0&0&0&0&1&-1&1&0&0\\0&0&0&0&0&-1&1&1&0\\0&0&0&0&0&1&-1&-1&0\\0&0&0&0&0&0&0&1&-1\\0&0&0&0&0&0&0&0&1\\\end{pmatrix}}


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


{\textbf  {v}}=(k_{{1}}S_{{0}}E_{1},k_{{-1}}C_{1},k_{{2}}C_{1},{\frac  {D_{{S}}}{h^{{2}}}}([S,-1]-[S,0],{\frac  {D_{{P_{1}}}}{h^{{2}}}}([P_{1},0]-[P_{1},+1]),k_{{3}}P_{{+1}}E_{2},k_{{-2}}C_{2},
k_{{4}}C_{2},{\frac  {D_{{P_{2}}}}{h^{{2}}}}([P_{2},+2]-[P_{2},+1]))


Considering the Reaction Space Only



S:D={\begin{pmatrix}-1&1&1&0&0&0&0&0&0\\1&-1&-1&0&0&0&0&0&0\\-1&1&0&1&0&0&0&0&0\\0&0&1&0&-1&0&0&0&0\\0&0&0&0&1&-1&1&0&0\\0&0&0&0&0&-1&1&1&0\\0&0&0&0&0&1&-1&-1&0\\0&0&0&0&0&0&0&1&-1\\\end{pmatrix}}


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


{\textbf  {v}}=(k_{{1}}S_{{0}}E_{1},k_{{-1}}C_{1},k_{{2}}C_{1},{\frac  {D_{{S}}}{h^{{2}}}}([S,-1]-[S,0],{\frac  {D_{{P_{1}}}}{h^{{2}}}}([P_{1},0]-[P_{1},+1]),k_{{3}}P_{{+1}}E_{2},k_{{-2}}C_{2},
k_{{4}}C_{2},{\frac  {D_{{P_{2}}}}{h^{{2}}}}([P_{2},+2]-[P_{2},+1]))


Right null space, rank r = 6

V_{{7}}=(-0.7342,-0.6386,-0.0965,-0.0965,-0.0965,-0.0094,0.0862,-0.0965,-0.0965)
V_{{8}}=(0.0120,0.2484,-0.2364,-0.2364,-0.2364,0.4434,0.6798,-0.2364,-0.2363)
V_{{9}}=(0.0502,-0.2678,0.3188,0.3188,0.3188,0.3188,0.5873,0.2685,0.3188,0.3188)