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 S_{1}, S_{3}, S_{4} and S_{7} 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 S_{1} to produce S_{3}, S_{4} and S_{7} 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 S_{1} and the maximum rate of the reaction respectively and K_{eq,n} is the thermodynamic equilibrium constant for the reaction S_{1} -> A S_{n} + B S_{7}. 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?

isa constant... Fixed now. – Chinmay Kanchi Apr 23 '10 at 1:07