Here in Copenhagen we’ll soon be having a bunch of interesting talks on chemical reaction networks:
• Workshop on Mathematical Trends in Reaction Network Theory, 1-3 July 2015, Department of Mathematical Sciences, University of Copenhagen. Organized by Elisenda Feliu and Carsten Wiuf.
Looking through the abstracts, here are a couple that strike me.
First of all, Gheorghe Craciun claims to have proved the biggest open conjecture in this field: the Global Attractor Conjecture!
• Gheorge Craciun, Toric differential inclusions and a proof of the global attractor conjecture.
This famous old conjecture says that for a certain class of chemical reactions, the ones coming from ‘complex balanced reaction networks’, the chemicals will approach equilibrium no matter what their initial concentrations are. Here’s what Craciun says:
Abstract. In a groundbreaking 1972 paper Fritz Horn and Roy Jackson showed that a complex balanced mass-action system must have a unique locally stable equilibrium within any compatibility class. In 1974 Horn conjectured that this equilibrium is a global attractor, i.e., all solutions in the same compatibility class must converge to this equilibrium. Later, this claim was called the Global Attractor Conjecture, and it was shown that it has remarkable implications for the dynamics of large classes of polynomial and power-law dynamical systems, even if they are not derived from mass-action kinetics. Several special cases of this conjecture have been proved during the last decade. We describe a proof of the conjecture in full generality. In particular, it will follow that all detailed balanced mass action systems and all deficiency zero mass-action systems have the global attractor property. We will also discuss some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.
Manoj Gopalkrishnan wrote a great post explaining the concept of complex balanced reaction network here on Azimuth, so if you want to understand the conjecture you could start there.
Even better, Manoj is talking here about a way to do statistical inference with chemistry! His talk is called ‘Statistical inference with a chemical soup’:
Abstract. The goal is to design an “intelligent chemical soup” that can do statistical inference. This may have niche technological applications in medicine and biological research, as well as provide fundamental insight into the workings of biochemical reaction pathways. As a first step towards our goal, we describe a scheme that exploits the remarkable mathematical similarity between log-linear models in statistics and chemical reaction networks. We present a simple scheme that encodes the information in a log-linear model as a chemical reaction network. Observed data is encoded as initial concentrations, and the equilibria of the corresponding mass-action system yield the maximum likelihood estimators. The simplicity of our scheme suggests that molecular environments, especially within cells, may be particularly well suited to performing statistical computations.
It’s based on this paper:
• Manoj Gopalkrishnan, A scheme for molecular computation of maximum likelihood estimators for log-linear models.
I’m not sure, but this idea may exploit existing analogies between the approach to equilibrium in chemistry, the approach to equilibrium in evolutionary game theory, and statistical inference. You may have read Marc Harper’s post about that stuff!
David Doty is giving a broader review of ‘Computation by (not about) chemistry’:
Abstract. The model of chemical reaction networks (CRNs) is extensively used throughout the natural sciences as a descriptive language for existing chemicals. If we instead think of CRNs as a programming language for describing artificially engineered chemicals, what sorts of computations are possible for these chemicals to achieve? The answer depends crucially on several formal choices:
1) Do we treat matter as infinitely divisible (real-valued concentrations) or atomic (integer-valued counts)?
2) How do we represent the input and output of the computation (e.g., Boolean presence or absence of species, positive numbers directly represented by counts/concentrations, positive and negative numbers represented indirectly by the difference between counts/concentrations of a pair of species)?
3) Do we assume mass-action rate laws (reaction rates proportional to reactant counts/concentrations) or do we insist the system works correctly under a broader class of rate laws?
The talk will survey several recent results and techniques. A primary goal of the talk is to convey the “programming perspective”: rather than asking “What does chemistry do?”, we want to understand “What could chemistry do?” as well as “What can chemistry provably not do?”
I’m really interested in chemical reaction networks that appear in biological systems, and there will be lots of talks about that. For example, Ovidiu Radulescu will talk about ‘Taming the complexity of biochemical networks through model reduction and tropical geometry’. Model reduction is the process of simplifying complicated models while preserving at least some of their good features. Tropical geometry is a cool version of algebraic geometry that uses the real numbers with minimization as addition and addition as multiplication. This number system underlies the principle of least action, or the principle of maximum energy. Here is Radulescu’s abstract:
Abstract. Biochemical networks are used as models of cellular physiology with diverse applications in biology and medicine. In the absence of objective criteria to detect essential features and prune secondary details, networks generated from data are too big and therefore out of the applicability of many mathematical tools for studying their dynamics and behavior under perturbations. However, under circumstances that we can generically denote by multi-scaleness, large biochemical networks can be approximated by smaller and simpler networks. Model reduction is a way to find these simpler models that can be more easily analyzed. We discuss several model reduction methods for biochemical networks with polynomial or rational rate functions and propose as their common denominator the notion of tropical equilibration, meaning finite intersection of tropical varieties in algebraic geometry. Using tropical methods, one can strongly reduce the number of variables and parameters of biochemical network. For multi-scale networks, these reductions are computed symbolically on orders of magnitude of parameters and variables, and are valid in wide domains of parameter and phase spaces.
I’m talking about the analogy between probabilities and quantum amplitudes, and how this makes chemistry analogous to particle physics. You can see two versions of my talk here, but I’ll be giving the ‘more advanced’ version, which is new:
Abstract. Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, the master equation for a chemical reaction network describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this formalism turns out to involve creation and annihilation operators, coherent states and other well-known ideas—but with a few big differences.
Anyway, there are a lot more talks, but if I don’t have breakfast and walk over to the math department, I’ll miss those talks!
You can learn more about individual talks in the comments here (see below) and also in Matteo Polettini’s blog:
• Matteo Polettini, Mathematical trends in reaction network theory: part 1 and part 2, Out of Equilibrium, 1 July 2015.