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Application to C5a Production with Spatial Component

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Applying Diffusion Approximation

We will apply our diffusion approximation method to the production of complement protein fragment C5a upon contact with a bacterial cell with diffusion across an extracellular capsule-plasma interface.

We know that the general form of the partial differential equations in the C5a production model is as follows:


{\frac  {\partial \phi }{\partial t}}=f(\phi )+D{\frac  {\partial ^{{2}}\phi }{\partial x^{{2}}}}.


Figure 1: Diagram of the points chosen for the approximation of diffusion. R1, R2, and R3 define the various distances separating regions in the spatial domain from each other.

We approximate diffusion by examining the behavior at a finite number of points, approximately evenly spaced along the spatial domain from the bacterial cell to the plasma, using the limit definition for the second derivative in accordance with Fick's Second Law, as derived previously. Specifically, to approximate the concentration change at a particular point x_{{i}}, we utilize information regarding the current concentration at three points: x_{{i-1}},x_{{i}},x_{{i+1}}. As we will see later, the boundary conditions inherent in this biological system will simplify some of these considerations. Since we are interested in developing a matrix for fundamental subspace analysis, we should theoretically only require three points in the spatial domain to obtain biologically accurate results. Point 1 will be defined as lying on the bacterial cell, Point 2 will be defined to be midway between the bacterial cell and the chosen capsule-plasma boundary, and Point 3 will lie on that capsule-plasma boundary. Of course, to utilize our chosen approximation method, we also require a "Point 0", lying somewhere in the interior of the bacterial cell, and "Point 4", lying out in the bloodstream of the system, beyond the spatial domain. (Figure 1) It is apparent that the concentrations at these points are irrelevant in our model, and as alluded to earlier, boundary conditions will eliminate them from our new system.

Given the boundary conditions in this model, our system of chemical "reactions" is

u_{{1}}\rightarrow products
u_{{1}}\leftrightarrow u_{{2}}
u_{{2}}\leftrightarrow u_{{3}}

The reaction essentially initiates upon contact with the bacterial cell surface, thus, our chemical reaction information (the stoichiometry of the system) will be contained entirely in the equations at point 1.

The last point we must consider before formulating the new diffusion-approximating model is the issue of boundary conditions. Observe that at the cell wall boundary, there is necessarily no diffusion in to or out of the cell. From our expression for the second derivative, we eliminate the "0 to 1 flux" found in the differential equation at point 1. Additionally, the concentration is uniform at the capsule-plasma boundary. This has an analogous result in our diffusion approximation equations, allowing us to eliminate the "4 to 3 flux" found in the differential equation at point 3. In a sense, this approximation blurs the line between point and domain, such that the "boundary" conditions are necessarily taken across an entire area: the Neumann condition in the space between points 0 and 1 and the Dirichlet condition in the space between 3 and 4.

Therefore, we have arrived at a general system of differential equations for diffusive components in this system at points 1, 2, and 3.


{\frac  {\partial \phi _{{1}}}{\partial t}}=f(\phi _{{1}})+D({\frac  {\phi _{{2}}-\phi _{{1}}}{h^{{2}}}})
{\frac  {\partial \phi _{{2}}}{\partial t}}=D({\frac  {(\phi _{{1}}-\phi _{{2}})+(\phi _{{3}}-\phi _{{2}})}{h^{{2}}}})
{\frac  {\partial \phi _{{3}}}{\partial t}}=D({\frac  {\phi _{{2}}-\phi _{{3}}}{h^{{2}}}})


These equations are effectively a decomposition of the original general equations and represent a spatial definition of both the stoichiometry and the diffusion of the system. Thus, the partial differential equations for the diffusive components of the system only can be approximated using these three equations. In our C5a production model, complement immunoproteins C3, C5, and C5a can be decomposed as such, but not cps and b, as they represent components of the system internal to the cell/capsule domain.

SVD Analysis of Diffusion-Approximating System

Explaining Our System of Differential Equations


Applying the general equations derived in the previous section to our C5a production interests should result in 11 differential equations -- five at the cell wall, three in the middle of the bacterial capsule, and three at the outer edge of the spatial domain, at the capsule-plasma boundary. They are as follows, labeled 1 for the bacterial cell wall, 2 for the middle of the capsule domain, and 3 for the outer boundary:


{\frac  {\partial [C3,1]}{\partial t}}=-K_{{C3}}[C3,1][b][cps]-K_{{tickover}}[C3,1]{\frac  {[cps]}{[cps_{{0}}]}}
+{\frac  {D_{{C3}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C3,2]-[C3,1])
{\frac  {\partial [cps]}{\partial t}}=-K_{{C3}}[C3,1][b][cps]-K_{{tickover}}[C3,1]{\frac  {[cps]}{[cps_{{0}}]}}
{\frac  {\partial [b]}{\partial t}}=K_{{C3}}[C3,1][b][cps]+K_{{tickover}}[C3,1]{\frac  {[cps]}{[cps_{{0}}]}}-K_{{decay}}[b]
{\frac  {\partial [C5,1]}{\partial t}}=-K_{{C5}}[b][C5,1]+{\frac  {D_{{C5}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C5,2]-[C5,1])
{\frac  {\partial [C5a,1]}{\partial t}}=K_{{C5}}[b][C5,1]+{\frac  {D_{{C5a}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C5a,2]-[C5a,1])


{\frac  {\partial [C3,2]}{\partial t}}={\frac  {D_{{C3}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}[([C3,1]-[C3,2])+([C3,3]-[C3,2])]
{\frac  {\partial [C5,2]}{\partial t}}={\frac  {D_{{C5}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}[([C5,1]-[C5,2])+([C5,3]-[C5,2])]
{\frac  {\partial [C5a,2]}{\partial t}}={\frac  {D_{{C5a}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}[([C5a,1]-[C5a,2])+([C5a,3]-[C5a,2])]


{\frac  {\partial [C3,3]}{\partial t}}={\frac  {D_{{C3}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C3,2]-[C3,3])
{\frac  {\partial [C5,3]}{\partial t}}={\frac  {D_{{C5}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C5,2]-[C5,3])
{\frac  {\partial [C5a,3]}{\partial t}}={\frac  {D_{{C5a}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C5a,2]-[C5a,3])


Of course, this necessarily alters the definition of our given initial concentrations (which are determined experimentally). Since the capsule and bound C3/C5 convertase remain "internal" to the cell/capsule domain, the initial conditions do not need to be redefined. However, the complement immunoproteins diffuse into contact with the bacterial cell and the cascade is initiated from our spatial domain. Therefore, we use our given initial condition quantities at the outer capsule-plasma boundary, or point 3, and set the initial conditions for the immunoproteins at points 1 and 2 to be equal to 0.

At this point, we can construct our stoichiometric/diffusion matrix for SVD analysis. It is logical to separate fluxes by diffusive gradient. In this situation, this is only applicable for the diffusion of immunoproteins through point 2 -- that is, {\frac  {D}{h^{{2}}}}(\phi _{{1}}-\phi _{{2}}) and {\frac  {D}{h^{{2}}}}(\phi _{{3}}-\phi _{{2}}), corresponding to the separate diffusion gradients on the "left hand side" and the "right hand side" of the spatial domains relative to point 2. It is worth noting that we are not explicitly specifying whether there is flux "into" or "out of" the system in terms of diffusion by Fick's First Law in our stoichiometric matrix. That is, if \phi _{{2}}>\phi _{{1}} (or \phi _{{3}}), a -1 should be entered into the stoichiometric matrix, but if \phi _{{2}}<\phi _{{1}} (or \phi _{{3}}), a +1 should be entered into the stoichiometric matrix, for the first hypothesized flux above. Still, as these are effectively equal but opposite fluxes, we can simply modify them for our biological situation: in this situation, we can represent both fluxes as +1's in the stoichiometric matrix, as the complement immunoproteins are flowing in from plasma into the extracellular capsule and into the cell surface, which have initial concentrations of 0.

As in the toy examples, observe that there are equal and opposite fluxes in our model, namely:


D({\frac  {\phi _{{2}}-\phi _{{1}}}{h^{{2}}}}) in {\frac  {\partial \phi _{{1}}}{\partial t}}=-D({\frac  {\phi _{{1}}-\phi _{{2}}}{h^{{2}}}}) in {\frac  {\partial \phi _{{2}}}{\partial t}}.
D({\frac  {\phi _{{3}}-\phi _{{2}}}{h^{{2}}}}) in {\frac  {\partial \phi _{{2}}}{\partial t}}=-D({\frac  {\phi _{{2}}-\phi _{{3}}}{h^{{2}}}}) in {\frac  {\partial \phi _{{3}}}{\partial t}}.


This eliminates a total of six fluxes {\textbf  {v}} from our model. Originally, relative to the "normal" model, we had twelve additional fluxes, corresponding to the diffusion of each complement immunoprotein at each of the three points, including separate "left and right" fluxes for each immune protein at point 2, but after noting the "equal but opposite" property under this approximation, we only have six additional fluxes. We also have six additional reactants, corresponding to creating differential equations for each complement immunoprotein at three points.

Thus, we have created an 11\times 10 matrix S:D such that


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


where


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


\phi =(cps,b,C3_{{1}},C3_{{2}},C3_{{3}},C5_{{1}},C5_{{2}},C5_{{3}},C5a_{{1}},C5a_{{2}},C5a_{{3}})


{\textbf  {v}}=(K_{{C3}}[C3][cps][b],K_{{tickover}}[C3]{\frac  {[cps]}{[cps_{{0}}]}},K_{{decay}}[b],K_{{C5}}[C5][b],
{\frac  {D_{{C3}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C3,2]-[C3,1]),{\frac  {D_{{C3}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C3,3]-[C3,2]),
{\frac  {D_{{C5}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C5,2]-[C5,1]),{\frac  {D_{{C5}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C5,3]-[C5,2]),
{\frac  {D_{{C5a}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C5a,2]-[C5a,1]),{\frac  {D_{{C5a}}}{[{\frac  {1}{2}}(R_{{2}}-R_{{1}})]^{{2}}}}([C5a,3]-[C5a,2])).



SVD Decomposition and Interpretation


Let m=11, corresponding to the rows of our matrix, and n=10, corresponding to the columns of our matrix.

We performed SVD analysis on this matrix using MATLAB's SVD command, generating three matrices U,\Sigma ,V^{{T}}, with U, where U is an m\times m unitary matrix, \Sigma is an m\times n diagonal matrix containing the singular values, and V^{{T}} is an n\times n unitary matrix. Furthermore, using MATLAB's rank operation for A gives a rank r=9.

Now, first, note that the dimension of the left null space is m-r=11-9=2, which means that the left null space has two spanning vectors. Through our SVD analysis, we see that the matrix U that an orthonormal basis for the left null sapce is composed of the vectors U_{{r+1}},...,U_{{m}}, which in this case is U_{{10}},U_{{11}}.


U_{{10}}=(-{\frac  {1}{2}},0,{\frac  {1}{2}},{\frac  {1}{2}},{\frac  {1}{2}},0,0,0,0,0,0)
U_{{11}}=(0,0,0,0,0,{\frac  {1}{{\sqrt  {6}}}},{\frac  {1}{{\sqrt  {6}}}},{\frac  {1}{{\sqrt  {6}}}},{\frac  {1}{{\sqrt  {6}}}},{\frac  {1}{{\sqrt  {6}}}},{\frac  {1}{{\sqrt  {6}}}})


U_{{10}},U_{{11}} correspond to conservation quantities in our model.

U_{{10}} corresponds to the conservation quantity C3_{{1}}+C3_{{2}}+C3_{{3}}-cps. By our model, this does indeed give a sum of differential equations equal to zero.
U_{{11}} corresponds to the conservation quantity C5_{{1}}+C5_{{2}}+C5_{{3}}+C5a_{{1}}+C5a_{{2}}+C5a_{{3}}. By our model, this also does give a sum of differential equations equal to zero.


We may need to modify the way we account for the capsule-plasma boundary according to this model, as these quantities should not really be time-invariant, unless you are assuming that all of the complement immunoproteins lie right at point 3 in our spatial domain at time t=0, which we effectively are doing by cancelling the flux \phi _{{4}}-\phi _{{3}}.

Meanwhile, the dimension of the right null space is n-r=10-9=1, meaning that the right null space has one spanning vectors. Indeed, through our SVD analysis, we see from taking the transpose of the V^{{T}} matrix (not reproduced here) that an orthonormal basis is composed of the vectors V_{{r+1}},...,V_{{n}}, or just V_{{10}} in this case:


V_{{10}}=({\frac  {1}{{\sqrt  {2}}}},-{\frac  {1}{{\sqrt  {2}}}},0,0,0,0,0,0,0,0)^{{T}}


Biologically, V_{{10}} corresponds to the steady-state flux distribution

K_{{C3}}[C3][cps][b]=-K_{{tickover}}[C3]{\frac  {[cps]}{[cps_{{0}}]}},

where the positive feedback reaction occurring for bound C3 to produce more C3 occurs at an equal and opposite rate to the tickover reaction, in which C3 entering the cell capsule binds to whatever binding spots are available within the spatial domain about the bacteria. All other fluxes must be held constant for this steady-state to arise: the decay reaction for bound C3 does not occur, C5 immunoprotein is not being cleaved into C5a complement fragment, and the complement components are not diffusing in and out of the spatial domain. Thus, unless the feedback and tickover reaction fluxes are zero, this biological situation is not realistic, and we conclude that there is no "steady-state" for this system, except for the trivial situation in which there no diffusion of complement immunoproteins and any complement components in the spatial domain are not binding to the capsule binding sites.

Infinite Points

As outlined in our toy examples, as the number of approximating points m\rightarrow \infty , we obtain a matrix of infinite rows and columns, as for each approximating point added, our diffusive matrix must account for a diffusive flux in to and out of one new point, generating an additional column, corresponding to an additional diffusive flux as well as an additional row, corresponding to an additional differential equation for an additional approximated point.

The capsule and bound C3 components remains independent of this process, because our differential equations for the capsule and the bound C3 do not have a spatial component. Additionally, the reaction stoichiometry itself remains the same, in that "point 1", or the point at the cell wall, is taken to be the site of immunological "action", with C3 binding to the bacterial capsule and C5 being cleaved into C5a fragment, and so forth.

From our S:D matrix written above for three points, we can remove the approximation by expanding the matrix in the same way that we saw in our toy examples. Note how the matrix is separated into "components": then the matrix will be expanded such that new "overlapping" columns are added for C3, C5, and C5a diffusions -- three expansions for each immunoprotein, as opposed to just one as seen in the toy example. Each row/column added will then add 1 to the matrix's rank, which suggests that the dimension of the left null space and the right null space will remain unchanged.

Fundamentally, the behavior of the left null space as m\rightarrow \infty is the same: namely, due to the overlapping nature of the diffusion from point to point, the total amount of C3 in the spatial domain subtracted by the total number of available capsule binding spots remainds constant (not listed in orthonormal form due to infinite points):

U=(1,0,1,...,1,0,0,0,0,0,0)

Also, the conservation relationship between the complement protein C5 and the protein fragment C5a remains as we remove the approximation.

U=(0,0,0,0,0,1,...,1)

The behavior of the right null space as m\rightarrow \infty is the same as well: based on the stoichiometry of the reaction, the only spanning vector in this space must be the biologically unrealistic (as discussed above) situation in which the positive feedback reaction occurring for bound C3 to produce more C3 occurs at an equal and opposite rate to the tickover reaction.

V=(1,-1,0,...,0)^{{T}}