Team:Brasil-USP/Modeling/EnzymaticReaction
Enzymatic Reactions
Modeling results
Table of contents
So far, we have modeled the production of enzymes that are capable of breaking rubber polymers, namely Lcp and RoxA. During the rubber degradation, however, no other products will be formed, only a bunch of small polymers with variable lengths. This means that, in a Michaelis-Menten fashion, our substrate and our product are the same and thus we would lose account of how much product we actually have. Then, we can ask which is the best way to model such behavior? Is Michaelis-Menten suitable? The result is partially yes, as we show below.
Figure 1- During the rubber degradation, Lcp and RoxA will break rubber polymers into smaller polymers. In a Michaelis-Menten fashion, our substrate and our product are the same. How can we solve it?
The main idea
This problem is not new, and in a similar way was studied before [1]. Since both our enzymes break polymers only in certain points, we can consider the substrate every breakable point in the system. See figure 2 to visualize this idea. In the figure, a polymer S has 3 breakable points. RoxA and Lcp will act on those points. This division separates monomeric units of size determined size (in our case, ODTD).
Figure 1- During the rubber degradation, Lcp and RoxA will break rubber polymers into smaller polymers. In a Michaelis-Menten fashion, our substrate and our product are the same. How can we solve it?
The only difference between Lcp and RoxA is simply how they break the polymer. RoxA has a kind of ruler: it always breaks at the lose ends of the polymer, generating one small piece and leaving the rest intact [2]. Lcp seems to break the polymer at random breakable points, creating a mixture of polymers with different sizes.
Since RoxA always breaks in the polymers in the same size and breaks the whole polymer at once [2], its dynamics can be modeled with a simple Michaelis-Menten. We will use this to find the proper kinetic for Lcp enzymatic reaction..
Dynamics of the Lcp enzymatic reaction
Supposing the validity of Michaelis-Menten with \(k_{cat}\) and \(k_m\) rate constants (as in figure 3) and suppose scission of the polymer at random (which is consistent with Lcp enzyme), then the degradation kinetics of the total number \(L(t)\) of breakable points at the instant \(t\) becomes
\[ \frac{dL}{dt} = - \frac{k_{cat} C_e L}{k_m + L}, \] where \(C_e\) is the concentration of the enzyme.
Figure 3- Michaelis-Menten dynamics.
Let \(m\) be the molecular weight of the monomeric unit and \(M\) the average weight of the chains, then \( N \left( \frac{M}{m} - 1 \right) \) is the cleavage limit of our reaction, with \(N\) being the total number of polymer.
This can be plugged into our gene expression models: \(C_e = P_{\ell} (t)\). With this final step, we can predict the full process bottom-up. Running enzymatic assays is not hard, however expressing Lcp and RoxA is still challenging for any research group. Unfortunately we could express nearly nothing of Lcp to run an assay and fit some parameters.
1- Cheng and Prud`Humme. Enzymatic Degradation of Guar and Substituted Guar Galactomannans, Biomacromolecules, 2000, vol 1, p.782.