Comments on: Network Theory (Part 24)

By: John Baez

John Baez — Mon, 10 Sep 2012 02:48:38 +0000

Thanks, that looks interesting! I hadn't seen it: • R. L. Karp, M. Pérez Millán, T. Dasgupta, A. Dickenstein and J. Gunawardena, Complex-linear invariants of biochemical networks.

Abstract: The nonlinearities found in molecular networks usually prevent mathematical analysis of network behaviour, which has largely been studied by numerical simulation. This can lead to difficult problems of parameter determination. However, molecular networks give rise, through mass-action kinetics, to polynomial dynamical systems, whose steady states are zeros of a set of polynomial equations. These equations may be analysed by algebraic methods, in which parameters are treated as symbolic expressions whose numerical values do not have to be known in advance. For instance, an "invariant" of a network is a polynomial expression on selected state variables that vanishes in any steady state. Invariants have been found that encode key network properties and that discriminate between different network structures. Although invariants may be calculated by computational algebraic methods, such as Gröobner bases, these become computationally infeasible for biologically realistic networks. Here, we exploit Chemical Reaction Network Theory (CRNT) to develop an efficient procedure for calculating invariants that are linear combinations of "complexes", or the monomials coming from mass action. We show how this procedure can be used in proving earlier results of Horn and Jackson and of Shinar and Feinberg for networks of deficiency at most one. We then apply our method to enzyme bifunctionality, including the bacterial EnvZ/OmpR osmolarity regulator and the mammalian 6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase glycolytic regulator, whose networks have deficiencies up to four. We show that bifunctionality leads to different forms of concentration control that are robust to changes in initial conditions or total amounts. Finally, we outline a systematic procedure for using complex-linear invariants to analyse molecular networks of any deficiency.

By: Cameron Smith

Cameron Smith — Sun, 09 Sep 2012 22:08:32 +0000

Perhaps you've already seen it, but it seems to me this paper may be relevant to this series or an adjacent one.

By: John Baez

John Baez — Sun, 02 Sep 2012 08:21:12 +0000

That looks important. One of her coauthors is Gavin Crooks, who has sometimes commented here, and who invented the wonderful Crooks fluctuation theorem. So, please remind me to learn about this, if I forget! Or even better, explain it to us.

By: Blake S. Pollard

Blake S. Pollard — Fri, 31 Aug 2012 23:50:39 +0000

This is a bit unrelated, but I just had the pleasure of attending a talk by Susanna Still, a Professor of Information and Computer Science out here at the University of Hawaii. The punchline of her talk was that for systems far from thermodynamic equilibrium (like most in nature) there is a “fundamental equivalence between … model inefficiency and thermodynamic inefficiency, measured by dissipation.” pointing to a deep connection ‘between the effective use of information and efficient thermodynamic operation.” Here is a link to a pre-print of her coming paper. http://arxiv.org/abs/1203.3271/

By: John Baez

John Baez — Thu, 23 Aug 2012 09:50:41 +0000

Thanks! I fixed the typo there.

Long time no see, Robert! Hope all’s well.

By: Robert Smart

Robert Smart — Thu, 23 Aug 2012 09:16:04 +0000

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