# [seqfan] Re: enumeration of siphons in chemical reaction networks

franktaw at netscape.net franktaw at netscape.net
Tue Jan 12 02:44:51 CET 2010

```Using the following PARI code:

v=vector(60);v[1]=1;for(k=4,60,v[k]=x*v[k-2]+x^2*v[k-3]);

I get

v[57] = 18*x^37 + 3876*x^36 + 77520*x^35 + 352716*x^34 + 497420*x^33 +
245157*x^32 + 42504*x^31 + 2300*x^30 + 26*x^29

which are the same coefficients in the paper, with an extra factor of
x^5.  This code gives the Padovan sequence for the coefficient sums.

Changing the initialization:

v=vector(60);v[1]=x^-1;v[2]=2;v[3]=1+x;for(k=4,60,v[k]=x*v[k-2]+x^2*v[k-3
]);

gives

v[50] = 18*x^32 + 3876*x^31 + 77520*x^30 + 352716*x^29 + 497420*x^28 +
245157*x^27 + 42504*x^26 + 2300*x^25 + 26*x^24

which exactly matches their values.

-----Original Message-----
From: franktaw at netscape.net

The total number is the Padovan sequence
(http://www.research.att.com/~njas/sequences/A000931), offset by 6.
The value 1221537 doesn't quite fit in the entry, but can be readily
found in the b-file.  A link to the paper should be added to that
sequence.

I'm not sure, from a brief glance, exactly what the component values
are, but yes, they should definitely be entered as a table, not just
the values for n=50.  Probably as a 'tabf', since otherwise it appears
there would be lots of zeros.

-----Original Message-----
From: Jonathan Post <jvospost3 at gmail.com>

26, 2300, 42504, 245157, 497420, 352716, 77520, 3876, 18

offset 24, 1

"The number of minimal siphons satisfies the recursion N(s) = N(s - 2)
+ N(s - 3), where N(2) = 2, N(3) = 2, and N(4) = 3. For s = 50 species
we
obtain N(50) = 1,221,537."

>From table on pp.15-16

Replacements for Wed, 6 Jan 10

http://arxiv.org/abs/0904.4529
Title: Siphons in chemical reaction networks
Authors: Anne Shiu, Bernd Sturmfels
Subjects: Commutative Algebra (math.AC); Molecular Networks
(q-bio.MN)

Abstract:
Siphons in a chemical reaction system are subsets of the species that
have the potential of being absent in a steady state. We present a
characterization of minimal siphons in terms of primary decomposition
of binomial ideals, we explore the underlying geometry, and we
demonstrate the effective computation of siphons using computer
algebra software. This leads to a new method for determining whether
given initial concentrations allow for various boundary steady states.

Or is this arbitrary (s = 50) and better presented as an array?

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```