2
$\begingroup$

From Michaelis-Menten kinetics one could easily derive scalar first-order differential equation $$\frac{dx}{dy} = A_0 + A_1 x + A_2 y + B \frac{y}{x},$$ where $A_0, A_1, A_2, B$ are some constants, that depend on rates of reactions.

I'm interested if there are any mathematical facts, that known for this type of equation. I doubt that it is possible to find explicit analytical solution in elementary functions. Also I see that this type of equation doesn't admit nonlinear superposition (unlike Riccati equation, for example).

$\endgroup$
1
  • $\begingroup$ I guess, we could re-write the scheme of reactions in a linear-algebraic style for initial condition and deal with eigenvectors/eigenvalues: en.wikipedia.org/wiki/Markov_chain (see "Chemistry"-> Michaelis-Menten on that page). $\endgroup$ Jun 18 '15 at 12:21
1
$\begingroup$

Maple classifies this as an Abel equation of type 2A. It can also be considered as a Chini equation with $n=-1$. The Chini invariant depends on $x$ (except in trivial cases: $A_0 = A_2 = 0$), so this does not lead to a closed-form solution. Moreover, Maple's symgen finds no symmetries.

There are some rather special cases with particular solutions. Thus $x = - A_2 y/A_1 $ is a solution if $B={\frac {{A_2}\, \left( {A_0}\,{ A_1}+{ A_2} \right) }{{{ A_1}}^{2}}} $

$\endgroup$
2
  • $\begingroup$ In quasisteady approximation (that is surely simplification of initial system of Michaelis-Menten) one has to involve Lambert-W function to get exact solution. I'm pretty sure without special function there is no chance to get closed-form solution for this equation... $\endgroup$
    – user47116
    Jun 19 '15 at 21:03
  • 1
    $\begingroup$ If LambertW was all that was needed to express a solution, Maple would be likely to find such a solution. But it does not. $\endgroup$ Jun 19 '15 at 21:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy