Analytical steady-state solution of a complex ODE

Question

I'm a biologist in the process of modeling a fairly simple biological system using a system of ODEs. To verify the simulations, I'm attempting to obtain an analytical steady-state solution that I can check the simulations against. My attempts so far haven't borne fruit, so I thought I'd toss the question out to mathematicians. This is my first post, so apologies if the question isn't right for this site.

The equation is of the form:

$${dS_3\over dt} = 2Xv_{max} {S_1 - {S_3^2S_7^4\over K_{eq,3}}\over K_m+S_1+{S_3^2S_7^4\over K_{eq,3}}} + D(S_{3,in} - S_3)$$

$${dS_4\over dt} = Xv_{max} {S_1 - {S_4S_7^2\over K_{eq,4}}\over K_m+S_1+{S_4S_7^2\over K_{eq,4}}} + D(S_{4,in} - S_4)$$

$${dS_1\over dt} = -Xv_{max} \Bigg[{S_1 - {S_3^2S_7^4\over K_{eq,3}}\over K_m+S_1+{S_3^2S_7^4\over K_{eq,3}}} + {S_1 - {S_4S_7^2\over K_{eq,4}}\over K_m+S_1+{S_4S_7^2\over K_{eq,4}}}\Bigg] + D(S_{1,in}-S_1)$$

$${dX\over dt} = Xv_{max}Y \Bigg[4{S_1 - {S_3^2S_7^4\over K_{eq,3}}\over K_m+S_1+{S_3^2S_7^4\over K_{eq,3}}} + 3{S_1 - {S_4S_7^2\over K_{eq,4}}\over K_m+S_1+{S_4S_7^2\over K_{eq,4}}}\Bigg] + D(X_{in}-X)$$

$${dS_7\over dt} = Xv_{max} \Bigg[4{S_1 - {S_3^2S_7^4\over K_{eq,3}}\over K_m+S_1+{S_3^2S_7^4\over K_{eq,3}}} + 2{S_1 - {S_4S_7^2\over K_{eq,4}}\over K_m+S_1+{S_4S_7^2\over K_{eq,4}}}\Bigg] + D(S_{7,in}-S_7)$$

Where S₁, S₃, S₄ and S₇ and X are variables

and

K_m, K_eq,3, K_eq,4, v_max, S_1,in, S_3,in, S_4,in, S_7,in, X_in, D and Y are constants.

This system models the change in the substrate S_n or the microbial population X in a perfectly-stirred vessel with microbes acting upon a substrate S₁ to produce S₃, S₄ and S₇ when the kinetics of the chemical reactions are thermodynamically reversible.

S_n,in is the input concentration of S_n. K_m and v_max are constants that describe the "affinity" of the microbe to S₁ and the maximum rate of the reaction respectively and K_eq,n is the thermodynamic equilibrium constant for the reaction S₁ -> A S_n + B S₇. I need to solve this system for S_n where n=1,3,4,7.

Is this even possible, or am I barking up the wrong tree here?

Answer 1

As Andrey mentioned, you shouldn't expect an analytical solution since you're dealing with a system of algebraic equation in several variables. (Just to be clear, what we're envisioning here is the system of equations you get by setting all the left-hand-sides to be zero). In your case, I believe you have 5 variables (X, $S_i$) and 5 equations, whose denominators can be cleared to make everything polynomial.

Such systems can be solved numerically (once you specify numerical values for your constants). One tool that I've used before is PHCpack, though for your purposes maybe Mathematica or something similar will be just fine.

Perhaps an expert can describe how to calculate resultants of your system of polynomials which will give you information on when the nature of the roots of this system change as you change your parameters...

Answer 2

Abbreviate $R_3={S_1 - {S_3^2S_7^4\over K_{eq,3}}\over K_m+S_1+{S_3^2S_7^4\over K_{eq,3}}}$ and $R_4={S_1 - {S_4S_7^2\over K_{eq,4}}\over K_m+S_1+{S_4S_7^2\over K_{eq,4}}}$. Then observing the positions of the $R$'s in your equations, set $C_1 = {1\over 2}S_3+S_4+S_1$, $C_2 = 2S_3+2S_4-S_7$ and $C_3=2YS_3+3YS_4-X$.

Then the $R$s cancel from your equations to give ${dC\over dt} = D(C_{in}-C)$ for each of $C_1$, $C_2$, and $C_3$. You can solve these explicitly, which ought to give you a good check.

In particular, if "input" means the same as "initial condition", then all three $C$'s are constants, conserved quantities.