Abstract
We are now in a position to revisit the mysteries described in Chapter 1. Recall that these were not mysteries about the behavior of chemical reaction networks themselves but, rather, about the behavior of people.
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Change history
02 February 2022
The original version of this book has been revised because it was inadvertently published with a few errors.
Notes
- 1.
It is presumed in this discussion that stoichiometric coefficients are integers.
- 2.
For example, the attempt to understand rich dynamical behavior in the cell cycle has been a particularly strong source of differential equations derived from experimentally inspired reaction network models. See, in particular, work by Tyson , Novak , Chen , and collaborators [32, 33, 167]. Recall too the discussion in Chapter 1 of the Huang-Ferrell network , formulated to engender differential equations governing the mitogen-activated protein kinase cascade [110, 137].
- 3.
By virtue of Theorem 10.6.14, the network is nondegenerate.
- 4.
In the sense of Section 4.2.2, the design parameters manifest themselves in rate constants for the feed and effluent reactions \(H2F \rightleftarrows 0\), \(NH \rightleftarrows 0\), H4F → 0, and N → 0.
- 5.
The chamber mixture is presumed to be perfectly stirred, and the concentrations of species in the effluent stream are presumed to be identical to their concentrations in the chamber.
- 6.
Computations were made with the help of XPP, software provided by Bard Ermentrout at http://www.math.pitt.edu/~bard/xpp/xpp.html.
- 7.
We will say more about the Deficiency One Algorithm in Part III.
- 8.
Throughout this subsection, we have in mind only networks with positively dependent reaction vectors, for those are the only ones that might admit a positive equilibrium or a periodic orbit containing a positive composition.
- 9.
Note, though, that a network with a discordant fully open extension must have a Species-Reaction Graph that violates at least one of the two conditions given in Theorem 11.6.1.
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Feinberg, M. (2019). The Big Picture Revisited. In: Foundations of Chemical Reaction Network Theory. Applied Mathematical Sciences, vol 202. Springer, Cham. https://doi.org/10.1007/978-3-030-03858-8_12
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