# Closed-form solution for this system of ODEs

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)...

• Isn't there supposed to be an $\dot{x}_a = \dots$ equation in there somewhere? It's underdetermined as it is... that being said, something that nonlinear is highly unlikely to admit a simple closed form solution... – gorilla Jul 12 '11 at 7:52
• @gorilla: you are right it is underdetermined, which is why I am looking for a closed-form solution (to be used afterward to do some statistical testing on potential parameter values). $\dot{x}_a$ does have a definition (similar to $\dot{x}_b$, but depending on some other $y_x$)... but it is not particularly helpful here... – Dave Jul 12 '11 at 8:03
• So, how did you manage to obtain a "Fourier series expansion" of $x_a$ if you don't have a "particularly helpful" definition of it? – gorilla Jul 12 '11 at 8:06
• @gorilla: while I do not have a "helpful definition" of $x_a$, I do have sampled values, which let me evaluate Fourier coefficients for a finite number of terms. I don't think this is even relevant, since the resolution part is done by simply assuming an unknown Fourier development for each of the three functions (then solving the system for a limited number of terms). Anyway, I am looking for an entirely different thing here... – Dave Jul 12 '11 at 8:25
• @gorilla: I should have probably specified that, while I do not have a useful explicit definition of $x_a$, I have some reasonably strong properties (bound, continuous etc.) as a result of it being a physical variable (concentration level of a chemical substance). Sorry if this wasn't clear... – Dave Jul 12 '11 at 8:29

## 1 Answer

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 $(y_a(s))_{s\le t}$.

• I must confess being curious to see how Fourier transform techniques can apply to differential equations involving Michaelis-Menten ratios. – Did Jul 12 '11 at 8:43
• Indeed... as I explained above, I am looking for a closed-form solution of this system in that I am hoping to ultimately express $x_b$ as a function of $x_a$ (through $y_a$). Thanks for your tip: this is more or less what I tried so far, but I must be missing something, because I do not see a way to use this form on the second equation (other than as a ratio of these two $u(t)$ forms, which does not look particularly friendly)... I am aware there is no particular reason for a simpler form to exist: just thought I'd ask... – Dave Jul 12 '11 at 8:49
• Perhaps I shouldn't have mentioned Fourier (expansion, not transform) above, as it seemed to have confused the question a bit, but its use allows me to reduce the ODEs above to a form where I can use my known estimate of Fourier coefficients for $x_a$ and $x_b$ to evaluate the remaining parameters. In essence solving analytically for my needs... – Dave Jul 12 '11 at 8:51
• Still curious to see Fourier expansions at work here. Another question: in real life people do consider exponents other than $n=1$? (Other than the trivial $n=0$, of course.) – Did Jul 12 '11 at 8:58
• Hard to go into too much details without being vastly off-topic, but there's really nothing fancy to it: just expressing all 3 functions as complex exponential sums (Fourier series) truncated to the first n terms, then using the above ODE to produce a system of equations in which known estimates for $\hat{x}_a(k)$ and $\hat{x}_b(k)$ can be substituted in order to evaluate the remaining parameters. Obviously, it accomplishes something much more limited than solving the above (but is enough for my purpose)... An analytical form would just be the cherry on top. :-) – Dave Jul 12 '11 at 9:04