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I am trying to solve the following system (derived from a Michaelis-Menten kinetics model for an enzymatic chemical reaction):

$\dot{y}_a = r_p x_a - \lambda_p y_a$

$\dot{x}_b = \frac{\alpha_0 + \alpha_1 (\frac{y_a}{K})^n}{ 1 + (\frac{y_a}{K})^n} - \lambda_m x_b$

Ideally, for all $n\in\mathbb Z$, but I would already be quite happy with $n \in \{-2, -1, 1, 2\}$

Currently, I use Fourier series expansions of $x_a$, $y_a$ and $x_b$ to rewrite the system and estimate the values I need...

I am wondering if there might be a closed-form solution to this system?

I think $y_a$ should be rewritable as an exponential function of $x_a$, but reinjecting this in the second equation got me nowhere (even when taking the $\log$... which straightens the fraction, but makes a mess of the rest).

I'd be really grateful for any pointer toward a closed-form solution (or indication that there is none)...

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    Also: thanks for that reference: I will check it out as soon as I can get my hands on it (unfortunately, our library system does not have access to it), but it is very likely to cover a different problem from mine. The model above is *derived* from Michaelis-Menten kinetics (applied to mRNA regulation), but significantly more complex, unfortunately.2011-07-13

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As explained by gorilla in the comments, you consider a system of two differential equations for three unknown functions hence you cannot hope for a solution. An auxiliary result which might help you though is as follows.

If $\dot u=v-cu$ then for every nonnegative $t$, $\displaystyle u(t)=\mathrm{e}^{-ct}\left(u(0)+\int_0^t\mathrm{e}^{cs}v(s)\mathrm{d}s\right).$

If one applies this to your first equation with $u=y_a$, $v=r_px_a$ and $c=\lambda_p$, one gets $y_a(t)$ as a function of $(x_a(s))_{s\le t}$. Likewise for the second equation and $x_b(t)$ as a function of $(y_a(s))_{s\le t}$.

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    actually, my bad: you can forget about negative values (they aren't particularly useful in this case). As for values of n > 1, they model cooperative/competitive models of chemical interactions common in gene regulation (Hill coefficient) and can generally go as high as 4 or 5.2011-07-13