September 24, 2017, Sunday
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Victoria's Research with Professor Bortz

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The focus of the research at hand has been to look into the kinetics of an enzyme substrate reaction from different perspectives. Initially, the following reaction was studied (where S is the concentration of the substrate, E is the concentration of the enzyme, SE is the concentration of the Substrate-Enzyme Complex, and P is the concentration of the product: S+E↔SE→E+P [1]

Figure 1: Substrate and Enzyme kinematics

Next, the code was modified to study the effects of an oscillation in one of the factors of the equation, on the other factors in the system. This was done by using the Lotka Volterra system as an oscillator, and adding this to the concentrations of either the substrate or enzyme:

x'=x(a-by) and y'=y-(c-dx) [2]

Whereas without any oscillations, the amount of enzyme and substrate stay at a relatively constant rate after they reach equilibrium—the substrate at zero and the enzyme at its initial concentration (Figure 1)—with the Lotka Volterra oscillations applied to the system, changes can be seen.

Figure 2: Substrate and Enzyme Oscillatory Inflow

In fact, by adding the oscillations, the effects of enzyme and substrate concentration, can be seen echoing through the system. Concluding from this, it can be seen that the substrate concentration has a much greater effect on the system kinematics as a whole than the enzyme concentration. In visual representation, this was clear because with oscillations in enzyme concentration, the amount of product, increased until it plateaued at equilibrium, and the enzyme concentration increased; whereas with oscillations in substrate concentration, the product concentration kept increasing. By adding these oscillations, attempts are made to model the behavior of the Complement Cascade. It works in such a way that the enzymes flow into the system and attach themselves to the pathogen in the bloodstream (the substrate). By inducing, within the system, oscillations in the inflow of substrate and enzyme, it is meant to incorporate the blood flow. As blood (oscillating with every heart beat) flows into the system, it brings more enzymes--which are assumed to be evenly distributed throughout the body--and more substrate--here assumed to be attached to the catheter or somehow else flowing in in a steady concentration with the heartbeat. Thus, it is reasonable to model them as flowing in, in an oscillatory fashion (Figure 2).

One of the greatest benefits of using a Lotka Volterra system to model the inflow of substance into the enzyme substrate system, is that it is possible to approximate the frequency of inflow using a linearized model of the Lotka Volterra system. To do this, the Jacobian of the Lotka Volterra system was found: Jacobian.jpg [3]

Here X_e and Y_e are the x and y positions of equilibrium of the model. With the use of this Jacobian, the frequency of oscillation of the linearized model was found to be √ac/(2*π). With the help of this newfound information, it was possible to see the effects of these oscillations on the substrate enzyme system. In order to take a more in-depth look at this effect, a Fourier Transform was done of the data. What this does is look at the frequencies at which the oscillations within the systems takes place. As expected, the frequencies would be the ones found using the Fourier Transform should be those calculated using equation 3, however because that is a linearized model, they were expected to deviate a little. Despite this deviation, graphing the Fourier Transform of the data confirmed past studies. According to this method of investigation, adding an oscillation to the enzyme concentration has a significantly greater effect on the product concentrations than adding to the substrate. This is likely because there is so much more substrate initially than enzyme, that mild fluctuations in the concentration of the substrate don’t change the product because all the enzymes are already “occupied with substrates” however adding more enzyme means an increase in the limiting reagent and thus more complexes can be formed, and out of those more products.

Hopefully, with this information and further studies, more can be learned about the workings of the Complement Cascade, especially looking at the way the enzyme and substrate concentrations, and the oscillations in these concentrations with the flowing of the blood, effect the rates at which pathogens can be cleared from the blood stream. By applying the simple enzyme substrate system currently being developed to each step in the behavior of the Complement Cascade, this will eventually be possible, and thus help gain a better understanding of the human Alternative Immune system response.