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SVD Analysis of C5a Production Through Bacterial Extracellular Polymeric Substance

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The immune complement is a complex network of enzymes, proteins, and receptors involved in the detection and destruction of invading organisms in the human body. One crucial component in this system is the complement immunoprotein fragment C5a, which is largely responsible for initiating the powerful host-defensive "membrane attack complex". Of course, there are consequences to the excessive production of various complement components, particularly C5a -- in particular, it is known to cause low blood pressure, cardiac dysfunction, and lymphocyte apoptosis. As a result, the production of this particular immune complement is particularly worth studying, as we will do in the following model.

Various previous efforts have revealed the enzyme kinetics of the complement system, which allows for the development of an accurate mathematical model. These efforts are valuable due to their revelation of behaviors in the system that occur at short distances and/or times that cannot be examined using current biological experimental techniques.

In conjunction with the University of Michigan Critical Care Engineering and Research Center, Biointerfaces Institute, and Department of Emergency Medicine, the Mathematical Biology group has produced a paper studying this topic in particular currently in review. In that particular work, these ideas are studied in context of a bloodstream infection arising from a bacterial infection (Staphylococcus epidermis) due to an indwelling medical device, such as a catheter.

We will utilize the same systems biology techniques seen earlier to analyze the reaction network of the complement activation stage of the immune reaction. The Law of Mass Action and previous understandings of complement diffusion, binding, and catalysis will assist us in formulating a mathematical model of differential equations. However, we extend our model to include diffusive terms -- that is, we wish to mathematically answer, "how does the presence of bacterial capsule material influence the kinetics of complement activation?". This changes the landscape of the system of differential equations significantly by introducing a spatial component, representing diffusion of complement materials to the bacterial cell surface.

Formulating the Model

In our model, we are interested in only five components: free C3 complement immunoprotein, free extracellular capsule (around the bacterial edge), C3/C5 bound to the extracellular capsule (in both active and inactive forms), free C5 complement immunoprotein, and free C5a complement immunoprotein. Additionally, we assume spherical symmetry, consistent with the general structure of S. epidermis. Regarding the diffusion of the system, then, the spatial domain is the the capsule, which is bounded at the inner edge by an impermeable cell wall and on the outer edge by the plasma domain of the bloodstream. These two factors gave rise to Dirichlet and Neumann boundary conditions at the boundaries of our spatial domain in the mathematical model. Another assumption made was that the diffusing and reacting species were not considered to be well-mixed, so spatial gradients were not considered.

Our model is then as follows:

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

where we assume uniform spherical symmetry, and thus \nabla ^{{2}}={\frac  {1}{r^{{2}}}}{\frac  {\partial }{\partial r}}(r^{{2}}{\frac  {\partial }{\partial r}}) in the above system.

Furthermore, we define the aforementioned boundary conditions as follows:

  • At the outer, capsule-plasma boundary is a Dirichlet boundary condition, where the reactant concentrations are equal to the concentrations in the capsule.
  • At the inner, cell wall boundary is a Neumann boundary condition, where there is zero flux, due to the impermeability of the bacterial wall.