# Mathematical Trends in Reaction Network Theory in Copenhagen /2

Following from here.

Ovidio Radulescu, Taming the complexity of biochemical networks through model reduction and tropical geometry. Actual Chemical Networks (CN) can be extremely complex, while rate equations cannot go much beyond 20 reactions. Hence one needs to find ways to reduce the model, by lumping reactions and replacing them by “elementary modes”. Most models have a multitude of time scales. Ovidio represents monomolecular reactions by integer-labelled digraphs. Rules for reduction are: “pooling” a fast cycle, “pruning” a fast reacton, etc. All of these are meant to somehow preserve the dynamics (while I have my own ideas on how to system-reduce basing on thermodynamics). So far though he was describing linear networks; the nonlinear case is much more complicated (although I did not understand in which respect his model reduction preserves the dynamics even in the linear case). The Tychonov-Penichel reduction theorems are mentioned again, which makes it one of the more important things I’ll have to learn about.

Nikki Meshkat, Algebraic techniques for the parameter identifiability problem in system biology. Suppose you have a state, an input function, an output function and a parameter. The question is which parameters can be identified given the input/output. One wants to be able to identify very clearly the parameter in a noise-free situation. She considers linear models, but all can be applied to nonlinear models. She considers a two-species model with an input current, and the output is the concentration of one of the two species. The only known thing is then a concentration and an influx. The unknown parameters of the problem are the kinetic rates. She needs to do some “differential elimination” (see Boulier 2007) to obtain a second-order equation for the input/output equation. Then from the time course she can estimate the coefficents of this second-order equation. And then the question is: is it possible to trace back the parameters, is the map from the parameters to the coefficients a 1-to-1 map? The map is polynomial, and most often it will be many-to-1, hence the system is “not identifiable”. Even in that case, though, she can come up with a reparametrization that maps the system into an identifiable system. Things are not always that easy, and the math becomes soon complicated, including Groebner basis, matroid theory (which seems to be more and more important in many areas…) etc.

Janos Toth, The form of kinetic differential equations. The first page was filled with formulas I could not understand.

David Doty, Computation by (not about) chemistry. This was the coolest talk, Doty is really good at captivating the attention by asking the audience to solve simple “exercises”. You may be wondering “why computing with chemistry?”, given that it’s slower and not significantly bigger. The reson might be that it adapts to wet hot environments. CRNs have many properties that might be suitable for computation: Boolean logic, be discrete or analog, have oscillations etc. He showed that many functions (linear functions, min and max) can be implemented by a chemical reaction, where some variable species $X_1,X_2$ etc. interact with “Yes” and “No” species; interaction between the species performs some computation (e.g. $f(X_1,X_2) = X_1-X_2$ is implemented by $X_2+Y \to \emptyset, X_1 \to Y$, interaction with Yes and Nos allows to program logical sentences and evaluate their truth value. Furthermore, CRNs can simuate counter machines (that are very slow universal Turing machines). This only holds if certain reactions can artificially be made extremely slow, and at very high energetic costs (all reactions are irreversible).

David Schnoerr. Breakdown of the chemical Langevin equation and moment closure approximations for stochastic chemical kinetics. Evidence for intrinsic noise in living cells has been found by Elowitz et al. Science 297. In E. Coli, genes are controlled by same promoters, hence extrinsic noise should be the same and if that was the only source of noise, then they should be perfectly correlated. Nevertheless the genes are expressed differently, hence there is intrinsic stochasticity. Scnoerr compared the Chemical Langevin equation and moment closure approximations to the Chemical Master Equation. The moment closure works like this: One takes the CME and obtains a hierarchy of moment equations, and then approximates higher-order moments in terms of lower-order moments, by assuming a distribution (e.g. Gaussian, which gives certain relations about the moments). Problems with negative CLE: molecule numbers can become negative! People tried to fix this problem; sometimes by posing a reflecting boundary at zero. Unfortunately this will not give the exact first two moments. A solution is to let the variables become complex; still the moments remain real, and this can be proven exactly, which is remarkable. Moreover, he showed by simulations that going complex performs much better in calculating mean and variance. Instead, moment closure approximations can fail to give the physical predictions, they diverge from the real values and can only be used above a certain critical volume. Interstingly, moment-closing to second moments is not equivalent to Van Kampen’s system-size expansion.

Daniele CappallettiComplex balanced reaction systems and Product-form Poisson distribution. He covered the great results relating complex-balance, deficiency and product-form Poisson-like steady distributions. Apparently some of the results can be stated in a more general setting than Mass-Action Law. Before, he gave me a very interesting example of a network that is not zero-deficiency but has product-form stationary distribution, namely $A+B \to 2A + B, 2A \to A$, which is formally product-form (because B does not evolve) but the rates differ in every stoichiometric compatibility class. The novelty is a definition of a “complex-balanced stationary distribution”, which is kind of a new definition. It turns out that it is always a product-form Poisson-like distribution, and that one can parallel for stochastic systems the same construction as Horn and Jackson and Feinberg, but for the Chemical Master Equation.