Simultanious DE - Kinetic Reaction

[paunonen89]

Hey, I'm looking for some help understanding how to solve these equations. The math is just a little bit beyond me. I would like to plot the equations over time to see how they are behaving.

These refer to the following equations:

Do I just sub all of these equations into number 14 to make a giant one to solve?

I have all the constants, the unknown values are G₁-G₇ and S₂,S₃,S₅. Also given are initial values for G_n/S₀, S_n/S₀ and V_max,n

The ultimate goal of this is to make a computer simulation, coded in vb.net. If you could also suggest a numerical integragtion method I can possibly turn it into a vb.net program that can solve the system...Thanks!

 

[yus310]

This problem is confusing in own right, but very interesting. Solving a system of differential equations simutaneously especially ones like these are complicated. The are a number of techniques, particularly using trapezoidal, simpson, runge kutta, or quadrature rules, but actually and realistically maybe just using the ode function in matlab, setting up the equations is a "function file" and then calling that function file in another main file, inputting the variables, etc. will do the trick.

Analytically by hand it is very difficult to integrate, and keep things in check, rather setting up a matrix for each of these solution of each equation, and then later plotting them vs. an arbitary X will do.

My suggestion is to first understand the correlation between the mathematics and physical phenomena, then to write the equations and the unknowns on a piece of paper, and make sure you copy everything down (knowns, unknowns, etc.). Then create a function file in matlab, inputting all these equations & tell the computer the variables (no numerical values in the equation). Then call the function in another file, and then input all the knowns, and then use the ode45 function to solve for each equation... and then finally you can plot each value vs. X, Gn, etc.

Give me sometime I will work on this too and get back to you. But hope this helps.

 

[paunonen89]

Here is a link to the full article.

http://www65.zippyshare.com/v/16801313/file.html

 

[yus310]

I will take a look at it.. sometime this week.. when is it due? and you are a grad. student?

 

[paunonen89]

I am an undergrad in Engineering. I am working on creating a model for my work term report.

 

[epenguin]

Intriguing. Saw this yesterday when you didn't have the downloads which I have not done - they look like they are music files!

As far as I can make out the model is of an amylase that digests short oligosaccharides to disaccharides (? this bit is confusing) and glucose which is then fermented to ethanol all in a vessel with no throughput.

You have something like 15 simultaneous equations! I cannot help with computers, but I imagine it is quite demanding even for computers. Or maybe not because each variable is connected with only one or two others.

You really know all of these K_ns and V_n's do you?

I would think the approach has to be see how far you can realistically simplify the model, see how that handles, then successively complexify it. If and when this does not change much you may decide there is no point in the more complex one. Actually even forget the 'realistically' at the start.



How can you tentatively simplify? Do you have any reason to think any of those S_n G_n are much larger (say > 4X) the corresponding K_m most of the time? I.e. enzyme saturated. Then you can approximate the corresponding Michaelian expression by a constant V_m. Or are any of them very much less than K_m,n? The expression can be simplified to V_max,nG_n/K_m,n for such case.

Then can you most of the time make a steady state approximation? E.g. dG₆/dt << dG₇/dt , making essentially dG₆/de = 0. Then you can reduce the number of variables in the d.e.

Play. Try this out for the top 3, say 7, 6, 5 - you might even be able to solve by hand.

Is it OK to open those files that look like music?

 

[paunonen89]

Zippyshare is the quickest and most straight forward free file hosting I have found. I just realized I can attach an article to a post so here it is.

 

[epenguin]

Not sure I understand all of this.

It seems the saccharfication part of the model is not affected by the rest of it.

Re what I said before about K_m,n's, I am not sure whether the Table 1 values express oligomer molarities or glucose equivalent molarities, I only guess the latter because then there is more of a trend. In any case they are some fraction of millmolar, so you seem to be at saturation all the time. I have some difficulty then in explaining Fig. 2 but maybe you are seeing saturation by some of the minor components. The experimental points are mostly above 3/4 saturation anyway. Then the concentrations in all the other experiments are higher still.

Maybe I'm wrong but I suspect you could treat the enzyme as saturated all the time which would be a quite significant simplification.

But then as the chains get short the k_max,n's decrease and thus the saccarification rate would too. But you would not go far wrong making all of 4-7 the same, average them, or better weight-average them according to their initial values. You could split glucose production into a process described by three, two or even one rate constants. Anyway if you assume saturation then equations are linear and you can solve the saccharifcation dynamics analytically by hand with even all 6 oligosaccharides with all their rate constants!

I do not understand eq. 7 and have never heard of an 'enzyme limitation constant'*. It looks like saturation of the substrate by enzyme, but I can't see any explanation and remain perplexed.

I therefore have reason to think there is a lot of overkill in this model. There are almost more parameters than experimental points in the saccarification part! (OK admittedly many independently obtained). Since you are just doing simulations for a short term project, a valid question for you to aim to answer might be 'How far can I reduce the complexity and still come up with adequate simulation of the experimental data?' Maybe the second part of the process, to which the first is a kind of independent input is different.

* I just googled it. The only mentions of it are from this paper and this thread!

 

[Redbelly98]

I would think the standard 4th-order Runge-Kutta method should solve this pretty easily. Just need to do the "grunt work", typing everything into set things up.

Or as yus310 said earlier, if you have access to MATLAB (or mathematica) then try that.