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Quantum Techniques for Studying Equilibrium in Chemical Reaction Networks

John C. Baez, Brendan Fong

Published May 21, 2013• 0 citations
OpenAlex

Abstract

Anderson, Craciun, and Kurtz have proved that a stochastically modelled chemical reaction system with mass-action kinetics admits a stationary distribution when the deterministic model of the same system with mass-action kinetics admits an equilibrium solution obeying a certain complex balance condition. Here we present a proof of their theorem using tools from the theory of second quantization: Fock space, annihilation and creation operators, and coherent states. This is an example of stochastic mechanics, where we take techniques from quantum mechanics and replace amplitudes by probabilities.

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Anderson, Craciun, and Kurtz have proved that a stochastically modelled chemical reaction system with mass-action kinetics admits a stationary distribution when the deterministic model of the same system with mass-action kinetics admits an equilibrium solution obeying a certain complex balance condition. Here we present a proof of their theorem using tools from the theory of second quantization: Fock space, annihilation and creation operators, and coherent states. This is an example of stochastic mechanics, where we take techniques from quantum mechanics and replace amplitudes by probabilities.
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of ‘chemical reaction networks’, which describes the interactions of molecules in a stochastic rather than quantum way. Computer science and population biology use the same ideas under a different name: ‘stochastic Petri nets’. But if we look at these theories from the perspective of quantum theory, they turn out to involve creation and annihilation operators, coherent states and other well-known ideas—but in a context where probabilities replace amplitudes. We explain this connection as part of a detailed analogy between quantum mechanics and stochastic mechanics. We use this analogy to present new proofs of two major results in the theory of chemical reaction networks: the deficiency zero theorem and the Anderson–Craciun–Kurtz theorem. We also study the overlap of quantum mechanics and stochastic mechanics, which involves Hamiltonians that can generate either unitary or stochastic time evolution. These Hamiltonians
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of "chemical reaction networks", which describes the interactions of molecules in a stochastic rather than quantum way. Computer science and population biology use the same ideas under a different name: "stochastic Petri nets". But if we look at these theories from the perspective of quantum theory, they turn out to involve creation and annihilation operators, coherent states and other well-known ideas - but in a context where probabilities replace amplitudes. We explain this connection as part of a detailed analogy between quantum mechanics and stochastic mechanics. We use this analogy to present new proofs of two major results in the theory of chemical reaction networks: the deficiency zero theorem and the Anderson-Craciun-Kurtz theorem. We also study the overlap of quantum mechanics and stochastic mechanics, which involves Hamiltonians that can generate either unitary or stochastic time evolution. These Hamiltonians are called "Dirichlet forms", and they arise naturally from electrical circuits made only of resistors.
Reaction networks are a general formalism for describing collections of classical entities interacting in a random way. While reaction networks are mainly studied by chemists, they are equivalent to Petri nets, which are used for similar purposes in computer science and biology. As noted by Doi and others, techniques from quantum physics, such as second quantization, can be adapted to apply to such systems. Here we use these techniques to study how the “master equation” describing stochastic time evolution for a reaction network is related to the “rate equation” describing the deterministic evolution of the expected number of particles of each species in the large-number limit. We show that the relation is especially strong when a solution of master equation is a “coherent state”, meaning that the numbers of entities of each kind are described by independent Poisson distributions. Remarkably, in this case the rate equation and master equation give the exact same formula for the time derivative of the expected number of particles of each species.
Relations between Hamiltonian mechanics and quantum mechanics are studied. It is stressed that classical mechanics possesses all the specific features of quantum theory: operators, complex variables, probabilities (in case of ergodic systems). The Planck constant and the Fock space appear after putting a dynamical system in a thermal bath. For harmonic oscillator in a thermal bath, the probability amplitudes can be identified with the complex valued phase functions f(q+ip)f(q+ip)f(q+ip) describing small deviations from the equilibrium state, when the time of relaxation is large. A chain of such oscillators models both the one-dimensional space (or string), and one-dimensional quantum field theory.
Near a detailed-balance equilibrium, the perturbed mass-action dynamics of a chemical reaction network (CRN) map exactly onto an electrical-flow problem on the bipartite species-reaction graph: chemical potentials become electrical potentials, Onsager coefficients become conductances, and the instantaneous Gibbs free-energy consumption equals the dissipated electrical energy. We exploit this map to design quantum walk algorithms that decide species reachability, sample reachable species, approximate any individual steady-state reaction flux, and estimate the total Gibbs dissipation. The first three follow from standard electrical-flow quantum walks; the last is non-trivial because the chemical flow is not the minimum-energy electrical flow on the same graph. We resolve this via a new use of alternative neighbourhoods in multidimensional quantum walks, which forces the walker onto the mass-action flow whenever the network is σ−Mσ-Mσ−M rigid. In an adjacency-matrix QRAM access model the algorithms achieve up to a quadratic speedup over classical methods -- for example Ω(n3/2)Ω(n^{3/2})Ω(n3/2) vs Ω(n2)Ω(n^2)Ω(n2) for reachability -- and dissipation-aware bounds tighten this further when the perturbation is concentrated.