I’ve seen Christoph Flamm, Daniel Merkle, Peter Sadler give talks on this project in Luxembourg, and it’s really fascinating. They’re using double pushout rewriting (as shown in the picture above) and other categorical techniques to design sequences of chemical reactions that accomplish desired tasks.

]]>• John Baez and Blake Pollard, A compositional framework for reaction networks, to appear in *Reviews in Mathematical Physics*.

But thanks to the arXiv, you don’t have to wait: *beat the rush, click and download now!*

Blake and I gave talks about this stuff in Luxembourg this June, at a nice conference called Dynamics, thermodynamics and information processing in chemical networks. So, if you’re the sort who prefers talk slides to big scary papers, you can look at those:

• John Baez, The mathematics of open reaction networks.

• Blake Pollard, Black-boxing open reaction networks.

]]>Here’s my attempt to formalize the idea. I fear I’m leaving out some important details.

A **reaction network** has a finite set of species , a finite set of reactions and source and target maps

The change in species due to a reaction is , and there is a linear map

usually called the **stochiometric matrix** defined by the equation

(Here I’m thinking of any element of as a linear combination of elements of , which form a basis; then I’m defining by its action on these basis elements.)

A **flux mode** is an element

for which

So, it’s a sum of reactions that, taken together, ‘don’t do anything’—it leaves the number of species of each sort unchanged.

An **elementary flux mode** is a flux mode that’s minimal, meaning there’s no nonzero flux mode with

Here I’m using the obvious partial order on flux modes, where iff for each species

I fear I’m leaving out some issues regarding enzymes: the reaction network I’m discussing might formally contain a reaction

but really ‘secretly’ this reaction could be

This wouldn’t significantly change the concept of flux mode, because the amount of the enzyme is unchanged, but in applications of flux modes we might want to know which flux modes are allowed if we put a constraint on how much of the enzyme is available.

Perhaps we should model this by a set of enzymes and a map

saying how much of each enzyme is involved in each reaction.

]]>If you count fatty acids with n carbon atoms, excluding ‘allenic’ ones (those with two neighboring double bonds, which are rare in nature) and ignoring cis/trans isomerism, you get the Fibonacci series!

]]>Schuster describes how elementary mode analysis has been used to study an interesting question: can fatty acids be transformed into glucose? Humans can’t do it using the reactions that convert glucose into fatty acids, but green plants, fungi, many bacteria (including *E. coli*) and nematodes can do it. That’s because they have the ‘glyoxylate shunt’.

Can humans convert fatty acids into sugar using some more complicated mechanism. Yes! – at least in principle. They’re very complicated: molecules need to cross the mitochondrial membrane three times! And yet, it seems they may be used by people playing soccer (for example), and by the Inuit, who eat a diet high in fat and low in sugar.

]]>Schuster defined an elementary mode of a chemical reaction network as “a minimal set of enzymes that can operate at steady state with all irreversible reactions used in the appropriate direction, with enzymes weighted by the relative flux they carry”. A more mathematical-sounding definition would help me a lot, and I’m sure one exists! I believe we’ve got a (directed multi)graph and we’re looking for a basis of the space of 1-chains obeying some property. There may be an assumption that all reactions involve an enzyme, meaning that they’re of the form

or perhaps

or

for some species the **enzyme**.

However they’re defined, the elementary modes are unique up to scaling, and they form a spanning set for some interesting vector space.

]]>• Stefan Schuster, Jan Ewald, Maximilian Fichtner, Severin Sasso and Christoph Kaleta, Modelling the link between lipid and carbohydrate metabolism in various species.

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AbstractElementary-modes analysis [1] has become a well-established theoretical tool in metabolic pathway analysis. It allows one to decompose complex metabolic networks into the smallest functional entities, which can be interpreted as biochemical pathways. It led to successful theoretical prediction of novel pathways, such as in carbohydrate metabolism inEscherichia coliand other bacteria [1, 2]. Metabolism is more complex than a graph in the sense of graph theory because of the presence of bimolecular reactions. Therefore, the existence of a connected route does not necessarily guarantee a net conversion along that route at steady state. This is here illustrated by tackling the question whether humans can convert fatty acids into sugar [3]. While, in agreement with biochemical dogma, no stoichiometrically balanced route for such a conversion can be found in human central metabolism, we did find several routes in a genome-scale network of human

metabolism [4]. This is likely to be relevant for sports physiology, weight-reducing diets and other applications. In green plants, fungi and many bacteria, in contrast, the above-mentioned conversion is enabled by the glyoxylate shunt, which is absent from humans and most animals. That shunt is of special importance for pathogenic fungi such asCandida albicans. Finally, we present a method for enumerating fatty acids [5], with potential applications in lipidomics and synthetic biology. We show that the number of unmodified fatty acids grows according to the famous Fibonacci numbers when cis/trans isomerism is neglected. Under consideration of that isomerism or modification by hydroxy- or oxo groups, diversity can be described by generalized Fibonacci numbers (e.g. Pell numbers).[1] S. Schuster, T. Dandekar, D.A. Fell, Detection of elementary flux modes in biochemical networks: A promising tool for pathway analysis and metabolic engineering,

Trends Biotechnol.17(1999) 53-60.[2] E. Fischer, U. Sauer: A novel metabolic cycle catalyzes glucose oxidation and anaplerosis in hungry

Escherichia coli, _J. Biol. Chem. 278 (2003) 46446–46451.[3] L. F. de Figueiredo, S. Schuster, C. Kaleta, D.A. Fell: Can sugars be produced from fatty acids? A test case for pathway analysis tools,

Bioinformatics25(2009) 152–158.[4] C. Kaleta, L.F. de Figueiredo, S. Werner, R. Guthke, M. Ristow, S. Schuster: In silico evidence for gluconeogenesis from fatty acids in humans,

PLoS Comp. Biol.7(2011) e1002116.[5] S. Schuster, M. Fichtner, S. Sasso: Use of Fibonacci numbers in lipidomics – Enumerating various classes of fatty acids,

Sci. Rep.7(2017) 39821.

It’s the first part of a two-part talk:

• Eric Smith and Supriya Krishnamurthy, Stochastic chemical reaction networks in the Doi-Peliti representation: scaling, moment hierarchies, and duality under the non-equiilbrium work relations and their generalizations.

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Abstract.The common material in our two talks will be the class of discrete-state stochastic processes associated with Chemical Reaction Networks (CRNs), as these are studied using the Doi-Peliti (DP) 2-field functional integral representation. The CRNs are an interesting class for which the graphical models representing the stochastic process are computationally complex, and the extent to which their solution properties can be determined from network topology is a problem of ongoing interest. The DP formalism gives a general way to represent generating functions and functionals for such processes, and provides insight into both solution properties of longstanding interest within the CRN community, and into the meaning of the duality of the Jarzynski/Crooks Non-Equilibrium Work Relations (NEWRs) and their generalizations.In the first segment, Eric Smith will review basics of CRN theory, including the similarities and differences to simpler diffusion processes, the sources of complexity, and the important topological characteristic known as “deficiency” due to Feinberg. He will show how representing stochastic CRNs with the DP formalism expresses simplicities and symmetries that are masked in more limited representations, and in particular how the duality associated with NEWRs is expressed. This talk will focus on the shift from causality to anti-causality that is generally associated with the NEWRs.

In the second segment, Supriya Krishnamurthy will show how the DP formalism combined with CRN theory helps in deriving a particularly simple representation of the hierarchy of moments. This representation provides a way to solve the moment hierarchies (in steady states) using matched asymptotic expansions. For some simple systems or for sub-hierarchies within more complex systems, the values of the moments can be solved from their asymptotic behaviours via direct numerical recursions.