Consequences of imperfect mixing the Gray-Scott model

Effects of intrinsic noise on a cubic autocatalytic reaction-diffusion system

Starting from our recent chemical master equation derivation of the model of an autocatalytic reaction-diffusion chemical system with reactions U+2V→[over λ_{0}]3V and V→[over μ]P, U→[over ν]Q, we determine the effects of intrinsic noise on the momentum-space behavior of its kinetic parameters and chemical concentrations. We demonstrate that the intrinsic noise induces n→n molecular interaction processes with n≥4, where n is the number of participating molecules of type U or V. The momentum dependences of the reaction rates are driven by the fact that the autocatalytic reaction (inelastic scattering) is renormalized through the existence of an arbitrary number of intermediate elastic scatterings, which can also be interpreted as the creation and subsequent decay of a three body composite state σ=ϕ_{u}ϕ_{v}^{2}, where ϕ_{i} corresponds to the fields representing the densities of U and V. Finally, we discuss the difference between representing σ as a composite or an elementary particle (molecule) with its own kinetic parameters. In one dimension, we find that while they show markedly different behavior in the short spatiotemporal scale, high-momentum (UV) limit, they are formally equivalent in the large spatiotemporal scale, low momentum (IR) regime. On the other hand, in two dimensions and greater, due to the effects of fluctuations, there is no way to experimentally distinguish between a fundamental and composite σ. Thus, in this regime, σ behaves as an entity unto itself, suggesting that it can be effectively treated as an independent chemical species.

Composite bound states and broken U(1) symmetry in the chemical-master-equation derivation of the Gray-Scott model

We give a first principles derivation of the stochastic partial differential equations that describe the chemical reactions of the Gray-Scott model (GS): U+2V →[λ]3V and V → [μ]P, U → [ν]Q, with a constant feed rate for U. We find that the conservation of probability ensured by the chemical master equation leads to a modification of the usual differential equations for the GS model, which now involves two composite fields and also intrinsic noise terms. One of the composites is ψ(1) = φ(v)(2), where {φ(v)}(η) =v is the concentration of the species V and the averaging is over the internal noise η(u,v,ψ(1)). The second composite field is the product of three fields χ = λφ(u)φ(v)(2) and requires a noise source to ensure probability conservation. A third composite ψ(2) = φ(u)φ(v) can also be identified from the noise-induced reactions. The Hamiltonian that governs the time evolution of the many-body wave function, associated with the master equation, has a broken U(1) symmetry related to particle number conservation. By expanding around the (broken symmetry) zero-energy solution of the Hamiltonian (by performing a Doi shift) one obtains from our path integral formulation the usual reaction diffusion equation, at the classical level. The Langevin equations that are derived from the chemical master equation have multiplicative noise sources for the density fields φ(u), φ(v),χ that induce higher-order processes such as n → n scattering for n>3. The amplitude of the noise acting on φ(v) is itself stochastic in nature.

Rate of mixing controls rate and outcome of autocatalytic processes: theory and microfluidic experiments with chemical reactions and blood coagulation

This article demonstrates that the rate of mixing can regulate the rate and outcome of both biological and nonbiological autocatalytic reaction systems that display a threshold response to the concentration of an activator. Plug-based microfluidics was used to control the timing of reactions, the rate of mixing, and surface chemistry in blood clotting and its chemical model. Initiation of clotting of human blood plasma required addition of a critical concentration of thrombin. Clotting could be prevented by rapid mixing when thrombin was added near the critical concentration, and mixing also affected the rate of clotting when thrombin was added at concentrations far above the critical concentration in two clinical clotting assays for human plasma. This phenomenon was modeled by a simple mechanism--local and global competition between the clotting reaction, which autocatalytically produces an activator, and mixing, which removes the activator. Numerical simulations showed that the Damköhler number, which describes this competition, predicts the effects of mixing. Many biological systems are controlled by thresholds, and these results shed light on the dynamics of these systems in the presence of spatial heterogeneities and provide simple guidelines for designing and interpreting experiments with such systems.