Reaction–diffusion processes and their connection with integrable quantum spin chains

For Abstract see ChemInform Abstract in Full Text.

We review the dynamics after quantum quenches in integrable quantum spin chains. We give a pedagogical introduction to relaxation in isolated quantum systems, and discuss the description of the steady state by (generalized) Gibbs ensembles. We then turn to general features in the time evolution of local observables after the quench, using a simple model of free fermions as an example. In the second part we present an overview of recent progress in describing quench dynamics in two key paradigms for quantum integrable models, the transverse field Ising chain and the anisotropic spin-1/2 Heisenberg chain.

The role of a reaction-diffusion system in the initiation of skin organ primordia. I. The first wave of initiation

Reaction-Diffusion (RD) systems provide a computational framework that governs many pattern formation processes in nature. Current RD system design practices boil down to trial-and-error parameter search. We propose a differentiable optimization method for learning the RD system parameters to perform example-based texture synthesis on a 2D plane. We do this by representing the RD system as a variant of Neural Cellular Automata and using task-specific differentiable loss functions. RD systems generated by our method exhibit robust, non-trivial 'life-like' behavior.

We construct integrable Hamiltonian systems on <f>$G/K$</f>, where <f>$G$</f> is a coboundary Poisson–Lie group and <f>$K$</f> is a Lie subgroup arising as the fixed point set of a group automorphism <f>$\\sigma $</f> of <f>$G$</f> satisfying the classical reflection equation. We show that the time evolution of these systems is described by a Lax equation, and under a factorizability assumption, present its solution in terms of a factorization problem in <f>$G$</f>. Our construction is closely related to the semiclassical limit of Sklyanin's integrable quantum spin chains with reflecting boundaries.Communicated by Anton Alekseev

Motivated by a study of the crossing symmetry of the asymmetric twin or ‘gemini’representation of the affine Hecke algebra we give a construction for crossing tensor spacerepresentations of ordinary Hecke algebras. These representations build solutions to theYang–Baxter equation satisfying the crossing condition (that is, integrable quantum spinchains). We show that every crossing representation of the Temperley–Lieb algebra appearsin this construction, and in particular that this construction builds new representations. Weextend these to new representations of the blob algebra, which build new solutions to theboundary Yang–Baxter equation (i.e. open spin chains with integrable boundaryconditions).We prove that the open spin chain Hamiltonian derived from Sklyanin’s commutingtransfer matrix using such a solution can always be expressed as the representation of anelement of the blob algebra, and determine this element. We determine the representationtheory (irreducible content) of the new representations and hence show that all suchHamiltonians have the same spectrum up to multiplicity, for any given value of thealgebraic boundary parameter. (A corollary is that our models have the samespectrum as the open XXZ chain with nondiagonal boundary—despite differingfrom this model in having reference states.) Using these multiplicity data, andother ideas, we investigate the underlying quantum group symmetry of the newHamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.

The stability of a one-spike solution to a general class of reaction-diffusion (RD) system with both regular and anomalous diffusion is analyzed. The method of matched asymptotic expansions is used to construct a one-spike equilibrium solution and to derive a nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O (1) time-scale. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined in terms of the roots of certain simple transcendental equations that involve two key parameters related to the choice of the nonlinear kinetics. From a rigorous analysis of these transcendental equations by using a winding number approach and explicit calculations, sufficient conditions are given to predict the occurrence of Hopf bifurcations of the one-spike solution. Our analysis determines explicitly the number of possible Hopf bifurcation points as well as providing analytical formulae for them. The analysis is implemented for the shadow limit of the RD system defined on a finite domain and for a one-spike solution of the RD system on the infinite line. The theory is illustrated for two specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation is unique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopf bifurcation.

Reaction–diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal–mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction–diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction–diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

Time delays, modelling the process of intracellular gene expression, have been shown to have important impacts on the dynamics of pattern formation in reaction–diffusion systems. In particular, past work has shown that such time delays can shrink the Turing space, thereby inhibiting patterns from forming across large ranges of parameters. Such delays can also increase the time taken for pattern formation even when Turing instabilities occur. Here, we consider reaction–diffusion models incorporating fixed or distributed time delays, modelling the underlying stochastic nature of gene expression dynamics, and analyse these through a systematic linear instability analysis and numerical simulations for several sets of different reaction kinetics. We find that even complicated distribution kernels (skewed Gaussian probability density functions) have little impact on the reaction–diffusion dynamics compared to fixed delays with the same mean delay. We show that the location of the delay terms in the model can lead to changes in the size of the Turing space (increasing or decreasing) as the mean time delay, tau , is increased. We show that the time to pattern formation from a perturbation of the homogeneous steady state scales linearly with tau , and conjecture that this is a general impact of time delay on reaction–diffusion dynamics, independent of the form of the kinetics or location of the delayed terms. Finally, we show that while initial and boundary conditions can influence these dynamics, particularly the time-to-pattern, the effects of delay appear robust under variations of initial and boundary data. Overall, our results help clarify the role of gene expression time delays in reaction–diffusion patterning, and suggest clear directions for further work in studying more realistic models of pattern formation.

We introduce a DNA-based reaction-diffusion (RD) system in which reaction and diffusion terms can be precisely and independently controlled. The effective diffusion coefficient of an individual reaction component, as we demonstrate on a traveling wave, can be reduced up to 2.7-fold using a self-assembled hydrodynamic drag. The intrinsic programmability of this RD system allows us to engineer, for the first time, orthogonal autocatalysts that counterpropagate with minimal interaction. Our results are in excellent quantitative agreement with predictions of the Fisher-Kolmogorov-Petrovskii-Piscunov model. These advances open the way for the rational engineering of pattern formation in pure chemical RD systems.

Integrable quantum spin chains display distinctive physical properties making them a laboratory to test and assess different states of matter. The study of the finite temperature properties is possible by use of the thermodynamic Bethe ansatz, however at the expense of dealing with non-linear integral equations for, in general, infinitely many auxiliary functions. The definition of an alternative finite set of auxiliary functions allowing for the complete description of their thermodynamic properties at finite temperature and fields has been elusive. Indeed, in the context of sln-invariant models satisfactory auxiliary functions have been established only for n≤4. In this paper we take a step further by proposing a systematic approach to generate finite sets of auxiliary functions for sln-invariant models. We refer to this construction as the canonical formulation. The numerical efficiency is illustrated for n=5, for which we present some of the thermodynamic properties of the corresponding spin chain.

By discretising space into compartments and letting system dynamics be governed by the reaction–diffusion master equation, it is possible to derive and simulate a stochastic model of reaction and diffusion on an arbitrary domain. However, there are many implementation choices involved in this process, such as the choice of discretisation and method of derivation of the diffusive jump rates, and it is not clear a priori how these affect model predictions. To shed light on this issue, in this work we explore how a variety of discretisations and methods for derivation of the diffusive jump rates affect the outputs of stochastic simulations of reaction–diffusion models, in particular using Turing’s model of pattern formation as a key example. We consider both static and uniformly growing domains and demonstrate that, while only minor differences are observed for simple reaction–diffusion systems, there can be vast differences in model predictions for systems that include complicated reaction kinetics, such as Turing’s diffusion-driven instability model of pattern formation. Our work highlights that care must be taken in using the reaction–diffusion master equation framework to make predictions as to the dynamics of stochastic reaction–diffusion systems.

A family of A2n(2) integrable open spin chains with Uq(Cn) symmetry was recently identified in arXiv:1702.01482. We identify here in a similar way a family of A2n−1(2) integrable open spin chains with Uq(Dn) symmetry, and two families of Dn+1(2) integrable open spin chains with Uq(Bn) symmetry. We discuss the consequences of these symmetries for the degeneracies and multiplicities of the spectrum. We propose Bethe ansatz solutions for two of these models, whose completeness we check numerically for small values of n and chain length N. We find formulas for the Dynkin labels in terms of the numbers of Bethe roots of each type, which are useful for determining the corresponding degeneracies. In an appendix, we briefly consider Dn+1(2) chains with other integrable boundary conditions, which do not have quantum group symmetry.

The interplay between reaction kinetics and diffusion leads to a wide range of spatiotemporal behaviors in reaction–diffusion (RD) systems. This article presents a theoretical and computational study of a RD system inspired by experimental observations of the Rho-GEF-Myosin signaling network controlling cell contraction dynamics. The temporal reaction system dynamics range from periodicity to bistability. By employing a dual hybrid dynamical systems approach of numerical bifurcation analysis and Floquet theory, we characterize the spatiotemporal dynamics when stable limit-cycle oscillators or homogeneous time-periodic solutions undergo the process of Floquet–Turing-diffusion-driven-instability (FTDDI). FTDDI refers to the emergence of spatially nonuniform patterns when diffusion destabilizes an otherwise temporally stable limit cycle of the underlying reaction kinetics. For the temporal reaction system, numerical bifurcation allows us to identify regions in a two-parameter space defining the stability of the uniform steady states and regions where the system exhibits limit cycles, which are characterized by employing the Floquet theory. In the presence of spatial variations, Floquet theory classifies regions where diffusion destabilizes the limit cycle or maintains their homogeneous stability and the emerging spatiotemporal dynamics of the full system, far from equilibrium. In the bistable regime, diffusion differentiates regions into those that exhibit classical Turing diffusion-driven instability, leading to pattern formation; and those that exhibit FTDDI, leading to space-time periodic patterns, spatially inhomogeneous patterns, oscillatory pulses, and wave propagation, and those that remain unaffected by diffusion. These findings provide theoretical insights into understanding complex spatiotemporal dynamics relevant to biological, chemical, and ecological spatiotemporal systems.

It is demonstrated that slowly travelling pulses arising in a reaction-diffusion (RD) system with the FitzHugh-Nagumo type nonlinearity do not necessarily annihilate but reflect off of each other before they collide. This phenomenon is in contrast with the well-known annihilation of travelling pulses on nerve axon and expanding rings in the Belousov-Zhabotinsky chemical reaction. By using singular perturbation methods, we derive a fourth order system of ODEs from the RD system, and study non-annihilation phenomenon of very slowly travelling pulses.