tldr;
Is there any benefit or disadvantage to writing the stoichiometry reactions of catalyzed by an enzyme as R + E --> P + E? Or is it better to write this R --> P? There's a graphical difference: is there a numerical one?
Details
I was recently implementing a very large (>300 species) in Simbiology. I was able to create the model using lists of compartments, parameters, species, and reactions. The graphics looked like a bowl of spaghetti. I made some progress on making sense of things but noted a bunch of dashed lines and wondered how those got into the model.
It turns out that if I have a reaction rate equation comprised of reactant, product, and enzyme (R, P, and E, respectively) with mass action rate expression (rate = kf * [R] * [E]) one way people write this is:
Stoichiometry: R + E --> P + E
As written, there's a dashed line on the model diagram between the Enzyme and the reaction. This was the case for the model I imported. In the diagram below this is case 2.
If one changes this so that we have:
Stoichiometry: R--> P
With the same rate expression, the dashed line disappears. This is shown as case 1. If you have a simple model the dashed lines might be useful. With 300 species, these dashed lines may not be what you want so much.
The issue is that adding the enzyme to both reactant and product lists then adds a bunch of zero-value differences to the rate equation for E. That is if E catalyzes a lot of reactions
dE/dt = r1 - r1 + r2 - r2 + r3 - r3 +..... + other stuff
Is there any reason that we shouldn't use the simpler R--> P stoichiometry to give a simpler diagram, and to remove useless calculations? I'm thinking that in integrating the model, there is little effect numerically of having all those (rate(i) - rate(i)) terms. Or perhaps having these terms mess up (makes less efficient) gradient calcs or sensitivity analysis?
As an aside, I would think that a mass action rate expression with E as both reactant and product might· be wrong in many cases in biology. Enzymes saturate. The case 2 stoichimetry might be thought to acknowledge this but it doesn't functionally address the issue. More properly, the reaction is probably more like case 3, below. Alternately Case 1 or Case 2, but with Michaelis-Menton kinetics rather than simple mass action. (MM is the pseudo-steady-state approximation for case 3, btw).
