
Multiple Bins (2)From MathBioIntroductionWe 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 stoichiometricdiffusion matrix for such a situation, as well as the characteristics of its fundamental subspaces. As an example, we will consider our open MichaelisMenten example in the context of bins.
Two MichaelisMenten Reaction BinsIn order to develop an understanding of the structure of a stoichiometricdiffusion matrix in the case of multiple bins, we begin by examining the simple case of two MichaelisMenten 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: . Note also that we assume the reaction and diffusion coefficients to be the same regardless of location. Chemically, we represent these identical reactions as:
<br.> Using our same diffusion approximation as a gradient, we obtain the following system of differential equations:
Using the same diffusion approximation as mentioned previously, we have generated a size matrix such that
,
where
As we have done previously, note that this method takes into account steadystates 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 stoichiometricdiffusion matrix (generating a matrix):
The rank of this matrix is , so the dimension of the right null space is , given by the following orthonormal basis:
We wish to extract biologically useful results from our fivedimensional 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 , it simply does not make sense for the and 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 , which produces
Combined with the diffusive fluxes, this situation actually looks fairly realistic. First, note the following flux balances: 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 MichaelisMenten system: namely, there must be constant at both points 0 and 1 in the domain.
n MichaelisMenten Reaction BinsWe 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 and 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 steadystate flux solution) has:
% Creates a MichaelisMenten stoichiometricdiffusion 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: MichaelisMenten stoichiometricdiffusion matrix %} function[mm_SD] = create_mm_sd(n) % Dimension of matrix must be 4n x 7n2 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,...,n1 for d = 1:1:(n1) mm_SD(d,3*n+1+d) = 1; mm_SD(d+1,3*n+1+d) = 1; end % for enzyme diffusion columns 1,...,n1 for e = 1:1:(n1) mm_SD(n+e,3*n+(n1)+1+e) = 1; mm_SD(n+e+1,3*n+(n1)+1+e) = 1; end % for complex diffusion columns 1,...,n1 for f = 1:1:(n1) mm_SD(2*n+f,3*n+2*(n1)+1+f) = 1; mm_SD(2*n+f+1,3*n+2*(n1)+1+f) = 1; end % for product diffusion columns 1,...,n1 for g = 1:1:(n1) mm_SD(3*n+g,3*n+3*(n1)+1+g) = 1; mm_SD(3*n+g+1,3*n+3*(n1)+1+g) = 1; end % for product diffusion column n mm_SD(4*n,7*n2) = 1;
