Re: From reactions to differential equations

• To: mathgroup at smc.vnet.net
• Subject: [mg93450] Re: From reactions to differential equations
• From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
• Date: Sat, 8 Nov 2008 04:00:50 -0500 (EST)

```Hi, Istvan,

just to add few words to the previous reply. I learned recently from the reply of M.Roellig that using Mathematica you can transform chemical equations into differential. This makes sense, if you deal with many tens or even hundreds of reactions. I rule out the evident problem related to stability of such a huge system. If one goes this way, he probably have successfully solved it.

However, if I understand you right, you asked another question: your problem is just to translate a certain (not too large) system of chemical reactions into differential equations and to analyze them. Right?

OK, but in this case it is an easy task, provided chemical reactions (like 2 Subscript[H, 2] + Subscript[O, 2] -> 2 Subscript[H, 2] O) are already established.

"...I would like to understand the method how these reactions are actually converted to diff.equations..."

I believe that your problem is not in Mathematica or any other program. I doubt that by use of any program one can understand such things. In contrast, the rules of translation a chemical reaction into a kinetic equation are explained in a number of books. Just in case you missed a good one, have a look into

Segel, L. A. Biological kinetics (eds. Cannings, C., Hoppenstedt, F. C. & Segel, L. A.) (Cambridge University Press, Cambridge, 1991).

I myself learned it first from one of Prigogine´s books, such as
Prigogine, Ilya (1961). Thermodynamics of Irreversible Processes, 2nd Ed, New York: Interscience  or
Prigogine, Ilya; Nicolis, G. (1977). Self-Organization in Non-Equilibrium Systems. Wiley. ISBN 0471024015.

There is of coarse, also a lot of other books on this subject. You need to read only few introductory chapters.

Even more, I would never trust any program to translate instead of me chemical reactions into differential equations. All the physical sense (chemical sense, biological sense - cross out those unnecessary) is hidden exactly in this step. It is here that you make assumptions and formulate your model. All the rest is some mathematical analysis which will never give you more than you put into your equations during this step.

I successfully used Mathematica several times to analyze equations describing chemical kinetics. However, if you have more than two-three non-linear equations describing your system, Mathematica may be not the best choice. I have seen at least one other than Mathematica program which was better designed for this purpose.  But moderator informed me that it is not eligible to give here its name.

Finally, I let myself one philosophic comment to the Community. In the MathGroup I often see questions which originated when the author cannot do something using Mathematica, but are in fact related to some lack of knowledge of the author in Mathematics or in some other discipline. So literally the answer does not belong to this forum. My attitude is that we should answer these questions nevertheless and give hints of where to look for the answer. Itself the fact that some question is related to some other field (rather than Mathematica) is something to do with Mathematica nevertheless. Just due to indication of boundaries.

Success, Alexei

> Hi Istvan,
>
> the reaction kinetic equations only tell you what is changed into what.
>
> But they do not give you the speed (rate constants). But the diff.
>
> equation are actually about speeds and you need the rate constants to
>
> write them.
>
> Further, Subscripts are only optical sugar, but they make programming
>
> hard. Therefore do not use it, or at least only for input/output.
>
> hope this helps, Daniel
>
>
>
> zac wrote:
>

> > > Dear All,
>
>

> > >
>
>

> > > from some time now I=E1m working with more and more complex differential
>
>

> > > equations.
>
>

> > > Is there a package to transform a set of reaction-kinetic equations
>
>

> > > such as:
>
>

> > >
>
>

> > > {
>
>

> > > Subscript[a, i] + Subscript[b, j] -> Subscript[c, i, j],
>
>

> > >  Subscript[a, i] + Subscript[c, i, j] -> 2 Subscript[a, i] +
>
>

> > > Subscript[b, j],
>
>

> > >  Subscript[b, i] + Subscript[c, j, i] -> 2 Subscript[b, i] +
>
>

> > > Subscript[c, j]
>
>

> > > }
>
>

> > >
>
>

> > > to differential equations expressing the growth of a_i, b_i and c_i,j,
>
>

> > > regarding
>
>

> > > that I leave the parameters i and j undefined?
>
>

> > > I am fully aware of software packages like COPASI and Systems Biology
>
>

> > > that are available, but I think they are far more complex than what is
>
>

> > > needed here, and furthermore I would like to understand the method how
>
>

> > > these reactions are actually converted to diff.equations. Which I
>
>

> > > completely miss if I'm to rely on a concealed algorithm.
>
>

> > > I've already started to write a parser in Mathematica, but it became
>
>

> > > too cumbersome after a point to deal with indexed variables (without
>
>

> > > indices it is fairly simple), and I really don't want to invent
>
>

> > > something that (perhaps) already exists on the field.
>
>

> > > Any ideas?
>
>

> > >
>
>

> > > Istvan Zachar
>
>

--
Alexei Boulbitch, Dr., Habil.
Senior Scientist

IEE S.A.
ZAE Weiergewan
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Luxembourg

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