Yeah, that gets into the technical details brushed under the rug. There’s two relevant types of equations governing the equilibrium value of PP:
The thermodynamic equilibrium equation, which fixes a product/ratio of the concentrations of the 3 species (linear in log-concentrations)
The stoichiometric constraints, which fix a couple linear combinations of the concentrations of the 3 species (linear in concentrations)
I’m effectively assuming that the thermodynamics favor X + Y over PP, so that the stoichiometric constraints can be approximated as “X and Y concentrations are each fixed”—there’s never enough PP produced to significantly decrease them. That way, we can ignore the stoichiometric limit (so long as X and Y are abundant), and just pay attention to the equilibrium equation. Then log-concentration of PP is a positive linear function of log-concentrations of X and Y, so everything is easy to think about. Increasing/decreasing either X or Y by enough can always shift the equilibrium PP above/below the threshold.
The problem with separate reactions (X → PP and Y → PP) is that, if Y is high, then increasing or decreasing X does nothing—PP will always be above threshold regardless of the X level. It’s an or-gate, rather than a linear function. Similarly, if PP were mainly determined by stoichiometric limits in my original reactions, we’d have an and-gate.
(I will need to think about it tomorrow, but are you effectively saying that the object of the experiment should be that thing Z which modifies what X+Y do?)
Yeah, that gets into the technical details brushed under the rug. There’s two relevant types of equations governing the equilibrium value of PP:
The thermodynamic equilibrium equation, which fixes a product/ratio of the concentrations of the 3 species (linear in log-concentrations)
The stoichiometric constraints, which fix a couple linear combinations of the concentrations of the 3 species (linear in concentrations)
I’m effectively assuming that the thermodynamics favor X + Y over PP, so that the stoichiometric constraints can be approximated as “X and Y concentrations are each fixed”—there’s never enough PP produced to significantly decrease them. That way, we can ignore the stoichiometric limit (so long as X and Y are abundant), and just pay attention to the equilibrium equation. Then log-concentration of PP is a positive linear function of log-concentrations of X and Y, so everything is easy to think about. Increasing/decreasing either X or Y by enough can always shift the equilibrium PP above/below the threshold.
The problem with separate reactions (X → PP and Y → PP) is that, if Y is high, then increasing or decreasing X does nothing—PP will always be above threshold regardless of the X level. It’s an or-gate, rather than a linear function. Similarly, if PP were mainly determined by stoichiometric limits in my original reactions, we’d have an and-gate.
(I will need to think about it tomorrow, but are you effectively saying that the object of the experiment should be that thing Z which modifies what X+Y do?)