• Dynamics, Thermodynamics and Information Processing in Chemical Networks, 13-16 June 2017, Complex Systems and Statistical Mechanics Group, University of Luxembourg. Organized by Massimiliano Esposito and Matteo Polettini.

They write, “The idea of the workshop is to bring in contact a small number of high-profile research groups working at the frontier between physics and biochemistry, with particular emphasis on the role of Chemical Networks.”

I’m looking forward to this, in part because there will be a mix of speakers I’ve met, speakers I know but haven’t met, and speakers I don’t know yet. I feel like reminiscing a bit, and I hope you’ll forgive me these reminiscences, since if you try the links you’ll get an introduction to the interface between computation and chemical reaction networks.

]]>It’s not that the yeast is doing something clever and complicated. If you break down the yeast cells, killing them, this effect still happens. People think these oscillations are inherent to the chemical reactions in glycolysis.

I learned this after writing Part 1, thanks to Alan Rendall. I first met Alan when we were both working on quantum gravity. But last summer I met him in Copenhagen, where we both attending the workshop Trends in reaction network theory. It turned out that now he’s deep into the mathematics of biochemistry, especially chemical oscillations!

]]>has **absolute concentration robustness** in , because this species has the same equilibrium value, *regardless of initial conditions!*

There are much more complicated models with this property, as shown in the paper by Feinberg and Shinar.

The theorem says: suppose you have a reaction network with a deficiency of one that admits a positive steady state. If, in the network, there are two nonterminal nodes that differ only in species S, then the system has absolute concentration robustness in S.

Amazing!

(In the above example the nodes and are nonterminal, since they have arrows pointing out of them, and they differ in one species.)

]]>• Martin Feinberg and Guy Shinar, Structural sources of robustness in biochemical reaction networks, *Science* **327** (2010), 1389–1391.

You can see a version of David’s talk here:

• David Anderson, Stochastic models of biochemical reaction systems: network structure and qualitative dynamics, 17 April 2015.

If you’re missing the prerequisites, start here:

• David Anderson, Tutorial: stochastic models of biochemical reaction systems, 16 April 2015.

His talk is based on this paper:

• David F. Anderson, Germán Enciso, and Matthew D. Johnston, Stochastic analysis of biochemical reaction networks with absolute concentration robustness, *Journal of the Royal Society Interface*, **11** (2014), 20130943.

• Vivien Brunel, Klaus Oerding and Frédéric van Wijland, Fermionic field theory for directed percolation in (1 + 1)-dimensions, *J. Phys. A: Math. Gen.* **33** (2000) 1085–1097.

I hope you’re familiar with the use of bosonic quantum field theory to study reaction-diffusion models, e.g. reviewed here:

• Johannes Knebel, *Application of Statistical Field Theory to Reaction-Diffusion Problems*, 29 April 2010.

MAPKs help cells in your body respond to a diverse array of stimuli. In a blog article, Rendall writes:

The MAP kinase cascade is a group of enzymes which can iteratively add phosphate groups to each other. More specifically, when a suitable number of phosphate groups have been added to one enzyme in the cascade it becomes activated and can add a phosphate to the next enzyme in the row. I found this kind of idea of enzymes modifying each other with the main purpose of activating each other fascinating when I first came across it.

Indeed, Alan’s talk reviewed a lot of interesting questions related to multistationarity, bistability and ‘biological clocks’ that show up chemical reaction network models of the MAP kinase cascade—and a lot of interesting answers, comingfrom both rigorous theorems and simulations.

You can see some of his work here:

• Alan Rendall and Juliette Hell, A proof of bistability for the dual futile cycle.

The **dual futile cycle** is this reaction network, which shows up as part of the MAP kinase cascade:

It’s called ‘futile’ because the phosphate groups, the P’s, get first added with the help of an enzyme E and then taken away with the help of an enzyme F. In this example we have at most two P’s, but we could have more. Then we’d have a **multiple futile cycle**.

In his paper Rendall writes:

The multiple futile cycle is an important building block in networks of chemical reactions arising in molecular biology. A typical process which it describes is the addition of n phosphate groups to a protein. It can be modelled by a system of ordinary differential equations depending on parameters. The special case is called the dual futile cycle.

There’s more in two papers of theirs that aren’t out yet, ‘Sustained oscillations in the MAPK cascade’ and ‘Dynamical features of the MAPK cascade’, and more on his blog:

• Alan Rendall, Dynamics of the MAP kinase cascade, *Hydrobates*, 7 April 2012.

• Alan Rendall, Proofs of dynamical properties of the MAPK cascade, *Hydrobates*, 3 April 2014.

The first one is the one I quoted.

]]>My impression is, her model is quite field theoretic. Colorful territories reminded me Ising (or Potts) model.

Besides, each lattice only admits one species. I thought this could be modeled by fermion…

Anyway, whether or not my idea is useful to describe her model, have you ever considered fermionic system, in your analogy? If you do, then could you give us some notes and readings in your blog or somewhere?

Below I would like to summarize what I have in mind.

Now the spectrum of the number operator for fermions is . Fock space may be constructed if you introduce Grassmann number. An element of the Fock space with species should look like

Creation and annihilation operators are of course multiplications and derivatives, and they satisfy anticommutation relation:

Now I have some troubles in constructing Hamiltonian and a master equation. Because I am using Grassmann number, for example the direct analogy does not seem to work…

I would be happy if you could give me any comments on these things!

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