Reaction-diffusion Theory Research Articles

Tuning nonequilibrium colloidal structure in external fields by density-dependent state switching.

Biological cells have the ability to switch internal states depending on the density of other cells in their local environment, referred to as "quorum sensing." The latter can be utilized to control collective structuring, such as in biofilm formation. In this work, we study a simple quorum sensing model of ideal (noninteracting) colloids with a switchable internal degree of freedom in the presence of external potentials. The colloids have two possible discrete states, in which they are affected differently by the external field, and switch with rates dependent on the local density in their environment. We study this model with reactive Brownian dynamics simulations, as well as with an appropriate reaction-diffusion theory. We find remarkable structuring in the system controlled by the density-mediated interactions between the ideal colloids. We report results of different functional forms for the rate dependence and quantify the influence of their parameters, in particular, discuss the role of the spatiotemporal sensing range, i.e., how the resulting structures depend on how the environmental information is "measured" by the colloids. Especially in the case of a rate function with sigmoidal dependence on local density, i.e., requiring a threshold density for switching, we observe significant correlation effects in the density profiles which are tuneable by the sensing ranges but also sensitive to noise and fluctuations. Hence, our model gives some basic insights into the nonequilibrium structuring mediated by simple quorum sensing protocols.

Theory of nanostructured sensors integrated in/on microneedles for diagnostics and therapy

Efficient real-time diagnostics and on-demand drug delivery are essential components in modern healthcare, especially for managing chronic diseases. The lack of a rapid and effective sensing and therapeutic system can result in analyte level deviations, leading to severe complications. Minimally invasive microneedle (MN)-based patches integrating nanostructures (NSs) in their volume or on their surface have emerged as a biocompatible technology for delay-free analyte sensing and therapy. However, a quantitative relationship for the signal response in NS-assisted reactions remains elusive. Existing generalized formalisms are derived for in-vitro applications, raising questions about their direct applicability to in-situ wearable sensors. In this study, we apply the reaction-diffusion theory to establish a generalized physics-guided framework for NS-in-MN platforms in wearable applications. The model relates the signal response to analyte concentration, incorporating geometric, physical, and catalytic platform properties. Approximating the model under NS (binding or catalytic) and environmental (mass transport) limitations, we validate it against numerical simulations and various experimental results from diverse conditions – analyte sensing (glucose, lactic acid, pyocyanin, miRNA, etc.) in artificial and in-vivo environments (humans, mice, pigs, plants, etc.) through electrochemical and optical/colorimetric, enzymatic and non-enzymatic platforms. The results plotted in the scaled response show that (a) NS-limited platforms exhibit a linear dependence, (b) Mass transport-limited platforms saturate to 1, (c) a one-to-one mapping against traditional sensitivity plots unifies the scattered data points reported in literature. The universality of the model provides insightful perspectives for the design and optimization of MN-based sensing technologies, with potential extensions to dissolvable MNs as part of analyte-responsive closed-loop therapeutic applications.

Turing Instabilities are Not Enough to Ensure Pattern Formation.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

A domain-dependent stability analysis of reaction–diffusion systems with linear cross-diffusion on circular domains

In this study, we present theoretical considerations of, and analyse, the effects of circular geometry on the stability analysis of semi-linear parabolic PDEs of reaction–diffusion type with linear cross-diffusion for a two-component system on circular domains. The highlights of our theoretical and computational findings are: (i) By employing linear stability analysis for a two-component reaction–diffusion system with linear cross-diffusion on circular disc domains, we derive necessary and sufficient conditions for the system to exhibit cross-diffusion driven-instability, dependent on the length scale of the geometry. These analytical studies involve cross-diffusion and circular geometry to unravel analytical conditions for the full computational classification of the parameter spaces that allow the system to exhibit Turing, Hopf and transcritical patterns. (ii) We compute parameter spaces on which patterns are formed only due to linear cross-diffusion as well as due to a critical domain length. These spaces do not exist in the absence of cross-diffusion nor when the conditions on the domain length are violated. (iii) To support our theoretical findings, finite element simulations illustrating the formation of spot patterns on circular domains are presented. Model parameter values are selected from parameter spaces that are induced by cross-diffusion, thereby supporting linear cross-diffusion coupled with reaction–diffusion theory as a candidate mechanism for pattern formation. (iv) A by-product of this study, is that an activator-depleted reaction–diffusion system with linear cross-diffusion on circular domains, appears to favour the formation of spot patterns for most of the parameter values chosen. Such patterns are reminiscent of those observed on stingrays, which form on approximately circular domains during growth development.

Evolution of intermetallics between solid Fe-Cr/Fe-Ni alloys and molten aluminium

Investigations of the evolution of intermetallic compounds (IMCs) at the interface between iron-based alloys and molten aluminium have important technical and scientific significance in many key areas. Most previous studies of IMCs formed at molten aluminium/solid steel interfaces have been focused on relatively long reaction times, but less research has been performed on the evolution of IMCs at the aluminizing interfaces of Fe-Cr and Fe-Ni alloys within a few seconds. In this study, hot-dip aluminizing experiments under natural convection conditions were employed to study the evolution of two interfacial IMCs (Fe-15Cr alloy and Fe-35Ni alloy with molten aluminium) at 700–900°C within 30 s. The IMCs for Fe-15Cr consisted of an Fe4Al13 layer next to the aluminium and an Fe2Al5 layer adjacent to the steel. In contrast, only one layer of Fe4Al13 was detected at the interface between the Fe-35Ni alloy and molten aluminium. The growth of Fe2Al5 was described by a parabolic rate law. An activation energy of 7.53 kJ/mol was determined for the temperature range of 700–900°C. The evolution of Fe4Al13 for the two alloys was similar: their thicknesses decreased anomalously with increasing dipping temperature, and their kinetics were controlled by both diffusion and dissolution. In this study, based on reaction-diffusion theory, growth and dissolution equations of the IMCs were established, and the diffusion coefficients of Al atoms in Fe2Al5 and Fe4Al13 were obtained for the first time. The similarities and differences in the IMCs for the two alloys were explained mainly by the different diffusion coefficients.

Turing/Turing-like patterns: Products of random aggregation of spatial components

Turing patterns are typical spatiotemporal ordered structures in various systems driven far from thermodynamic equilibrium. Turing’s reaction-diffusion theory, containing a long-range inhibiting agent and a local catalytic agent, has provided an explanation for the formation of some patterns in nature. Numerical, experimental and theoretical studies about Turing/Turing-like patterns have been generally focused on systems driven far from thermodynamic equilibrium. The local dynamics of these systems are commonly very complex, which brings great difficulties to understanding of formation of patterns. Here, we investigate a type of Turing-like patterns in a near-equilibrium thermodynamic system experimentally and theoretically, and put forward a new formation mechanism and a quantitative method for Turing/Turing-like patterns. Specifically, we observe a type of Turing-like patterns in starch solutions, and study the effect of concentration on the structure of patterns. The experimental results show that, with the increase of concentration, patterns change from spots to inverse spots, and labyrinthine stripe patterns appear in the region of intermediate concentration. We analyze and model the formation mechanism of these patterns observed in experiments, and the simulation results agree with the experimental results. Our conclusion indicates that the random aggregation of spatial components leads to formation of these patterns, and the proportion of spatial components determines the structures. Our findings shed light on the formation mechanism for Turing/Turing-like patterns.

Large self-assembled clathrin lattices spontaneously disassemble without sufficient adaptor proteins.

Clathrin-coated structures must assemble on cell membranes to internalize receptors, with the clathrin protein only linked to the membrane via adaptor proteins. These structures can grow surprisingly large, containing over 20 clathrin, yet they often fail to form productive vesicles, instead aborting and disassembling. We show that clathrin structures of this size can both form and disassemble spontaneously when adaptor protein availability is low, despite high abundance of clathrin. Here, we combine recent in vitro kinetic measurements with microscopic reaction-diffusion simulations and theory to differentiate mechanisms of stable vs unstable clathrin assembly on membranes. While in vitro conditions drive assembly of robust, stable lattices, we show that concentrations, geometry, and dimensional reduction in physiologic-like conditions do not support nucleation if only the key adaptor AP-2 is included, due to its insufficient abundance. Nucleation requires a stoichiometry of adaptor to clathrin that exceeds 1:1, meaning additional adaptor types are necessary to form lattices successfully and efficiently. We show that the critical nucleus contains ~25 clathrin, remarkably similar to sizes of the transient and abortive structures observed in vivo. Lastly, we quantify the cost of bending the membrane under our curved clathrin lattices using a continuum membrane model. We find that the cost of bending the membrane could be largely offset by the energetic benefit of forming curved rather than flat structures, with numbers comparable to experiments. Our model predicts how adaptor density can tune clathrin-coated structures from the transient to the stable, showing that active energy consumption is therefore not required for lattice disassembly or remodeling during growth, which is a critical advance towards predicting productive vesicle formation.

Mechanism and performance relevance of nanomorphogenesis in polyamide films revealed by quantitative 3D imaging and machine learning.

Biological morphogenesis has inspired many efficient strategies to diversify material structure and functionality using a fixed set of components. However, implementation of morphogenesis concepts to design soft nanomaterials is underexplored. Here, we study nanomorphogenesis in the form of the three-dimensional (3D) crumpling of polyamide membranes used for commercial molecular separation, through an unprecedented integration of electron tomography, reaction-diffusion theory, machine learning (ML), and liquid-phase atomic force microscopy. 3D tomograms show that the spatial arrangement of crumples scales with monomer concentrations in a form quantitatively consistent with a Turing instability. Membrane microenvironments quantified from the nanomorphologies of crumples are combined with the Spiegler-Kedem model to accurately predict methanol permeance. ML classifies vastly heterogeneous crumples into just four morphology groups, exhibiting distinct mechanical properties. Our work forges quantitative links between synthesis and performance in polymer thin films, which can be applicable to diverse soft nanomaterials.

Large self-assembled clathrin lattices spontaneously disassemble without sufficient adaptor proteins

Clathrin-coated structures must assemble on cell membranes to internalize receptors. These structures can grow surprisingly large, containing over 20 clathrin seemingly locked in a hexagonal lattice, yet often fail to form productive vesicles. We show that these abortive events happen spontaneously when adaptor availability is low, despite abundant clathrin. Here, we combine recent in vitro kinetic measurements with stochastic reaction-diffusion simulations and theory to differentiate mechanisms of stable vs unstable clathrin assembly on membranes.

Modern perspectives on near-equilibrium analysis of Turing systems.

In the nearly seven decades since the publication of Alan Turing’s work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction–diffusion theory. Some of these developments were nascent in Turing’s paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction–diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of ‘trivial’ base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality.This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Pattern formation in Passiflora incarnata: An activator-inhibitor model

Based on a careful examination of the onset of violet colored dots along the filaments in the developing floral bud stage and the formation of alternating bands of violet and white color in the matured flowers of Passiflora incarnata (Passion flower), it is concluded that the pattern arises from a competition between the production of violet colored anthocyanin and the colorless flavonols along the filaments. The activator-inhibitor model of Gierer and Meinhardt along with the reaction diffusion theory of Turing is used to explain the formation of concentric rings in the flower.

Nanoscale Turing patterns in a bismuth monolayer

Turing’s reaction–diffusion theory of morphogenesis has been very successful for understanding macroscopic patterns within complex objects ranging from biological systems to sand dunes. However, Turing patterns on microscopic length scales are extremely rare. Here we show that a strained atomic bismuth monolayer assembled on the surface of NbSe2—and subject to interatomic interactions and kinetics—displays Turing patterns. Our reaction–diffusion model produces stripe patterns with a period of five atoms (approximately 2 nm) and domain walls with Y-shaped junctions that bear a striking resemblance to what has been experimentally observed. Our work establishes that Turing patterns can occur at the atomic scale in a hard condensed-matter setting.

Field theory of reaction-diffusion: Law of mass action with an energetic variational approach.

We extend the energetic variational approach so it can be applied to a chemical reaction system with general mass action kinetics. Our approach starts with an energy-dissipation law. We show that the chemical equilibrium is determined by the choice of the free energy and the dynamics of the chemical reaction is determined by the choice of the dissipation. This approach enables us to couple chemical reactions with other effects, such as diffusion and drift in an electric field. As an illustration, we apply our approach to a nonequilibrium reaction-diffusion system in a specific but canonical setup. We show by numerical simulations that the input-output relation of such a system depends on the choice of the dissipation.

A unified aging compact model for hot carrier degradation under mixed-mode and reverse E-B stress in complementary SiGe HBTs

This paper presents an accurate, comprehensive and physics-based aging compact model for stress-induced degradation due to hot-carrier generation and oxide trapping in advanced complementary NPN and PNP SiGe HBTs. The analytical model equations are derived from the solution of reaction–diffusion theory and Fick’s law of diffusion combined with oxide trapping mechanism under accelerated stress conditions. The model accuracy has been validated against results from long-term aging tests performed close to the safe-operating-areas of an advanced complementary 0.25 µm BiCMOS technology. Degradation asymmetry observed between NPN and PNP devices is accurately captured by this unified aging compact model. This study highlights the challenges of predicting degradation of complementary circuits and thereby improving its functionalities by designing better-matched NPN and PNP HBTs.

Mass Transfer Modeling of α‐Eleostearic Acid from Tung Oil Concentration by Low‐Temperature Crystallization

AbstractWith the significant attention paid to α‐eleostearic acid (α‐ESA) in a variety of biologically beneficial effects, the demand for high‐concentration products is increasing. A method of cryogenic separation to predict the mass variation of α‐ESA from tung oil was established in this study. The results showed that the best α‐ESA crystallization condition was −20 °C in ethanol, which implied a low energy consumption when compared with the common crystallization temperature of −60 °C. The α‐ESA concentration in the liquid phase was reduced by 90.4% compared to tung oil, and most of the α‐ESA in the solution was recovered by mass transfer. On the other hand, a mathematical model based on the diffusion reaction theory is used in mass variation prediction. The effects of temperature and time on the mass variation in α‐ESA in solution crystallization were examined. The results showed that the proposed model could estimate the composition and total mass of winterized tung oil at R2=0.974. The model was validated by experiments with an error of ±10%.

A physical and versatile aging compact model for hot carrier degradation in SiGe HBTs under dynamic operating conditions

This paper presents a new physics-based compact model implementation for interface state creation due to hot-carrier degradation in advanced SiGe HBTs. This model accounts for dynamic stress bias conditions through a combination of the solution of reaction-diffusion theory and Fick’s law of diffusion. The model reflects transistor degradation in terms of base recombination current parameters of HiCuM compact model and its accuracy has been validated against results from long-term DC and dynamic aging tests performed close to the safe-operating-areas of various HBT technologies.

151. Hyperthermia, generated by focused ultrasound, in combination with radiations for a more effective glioblastoma therapy

Purpose The existing transcranial focused ultrasound device (Insightec Exablate 4000 Neuro) can induce hyperthermia in a brain region ten times larger than was recently covered with ablation [1] . The purpose of this work was to theoretically derive how a relatively small volume of glioblastoma multiforme (GBM) can be effectively treated by combining focused ultrasound based hyperthermia with an external beam radiotherapy. Methods We applied a mathematical descriptions of GBM proliferation and diffusion, in the frame of the reaction diffusion theory. We formulated equations capturing of the impact of radiotherapy and heat on GBM in the reaction diffusion equation, including the regrowth induced by stem cells [2] . Results Our calculated results show that the expected patient survival could be significantly improved by this treatment [3] . For the future, due to the lower power required for a single heating point, a somewhat lower frequency could be hypothesized, without risks of cavitation. This modification would allow the heating of larger tumors in less central regions. Conclusions The proposed methodology, now limited to a small tumors in a quite central location, can be easily improved in several different respects. These results advocate thorough experiments to validate this very promising theoretical therapeutic benefit.

Analysis of Transmission of Infection in Epidemics: Confined Random Walkers in Dimensions Higher Than One.

The process of transmission of infection in epidemics is analyzed by studying a pair of random walkers, the motion of each of which in two dimensions is confined spatially by the action of a quadratic potential centered at different locations for the two walks. The walkers are animals such as rodents in considerations of the Hantavirus epidemic, infected or susceptible. In this reaction-diffusion study, the reaction is the transmission of infection, and the confining potential represents the tendency of the animals to stay in the neighborhood of their home range centers. Calculations are based on a recently developed formalism (Kenkre and Sugaya in Bull Math Biol 76:3016-3027, 2014) structured around analytic solutions of a Smoluchowski equation and one of its aims is the resolution of peculiar but well-known problems of reaction-diffusion theory in two dimensions. The resolution is essential to a realistic application to field observations because the terrain over which the rodents move is best represented as a 2-d landscape. In the present analysis, reaction occurs not at points but in spatial regions of dimensions larger than 0. The analysis uncovers interesting nonintuitive phenomena one of which is similar to that encountered in the one-dimensional analysis given in the quoted article, and another specific to the fact that the reaction region is spatially extended. The analysis additionally provides a realistic description of observations on animals transmitting infection while moving on what is effectively a two-dimensional landscape. Along with the general formalism and explicit one-dimensional analysis given in Kenkre and Sugaya (2014), the present work forms a model calculational tool for the analysis for the transmission of infection in dilute systems.

Analytical Low Frequency NBTI Compact Modeling with H2 Locking and Electron Fast Capture and Emission

Besides reaction-diffusion theory explaining the generation and passivation of interface trap (ΔNIT), hole trapping/de-trapping in preexisting gate insulator traps and transient charge occupancy in ΔNIT are also combined to describe the characteristic of NBTI degradation. However, it is found that H2 locking effect and Electron Fast Capture/Emission play key roles in the NBTI degradation. In this paper, an analytical low frequency AC NBTI compact model has been proposed to accurately predict the shift in threshold voltage. Two fitting parameters (α and FFAST) have been introduced to account for the H2 locking and fast electron capture and emission. The comparison between the proposed model and the experimental data has been carried out, and the results show that our proposed can catch the kinetics of NBTI degradation under low frequency AC stress conditions.

Mechanics of self-healing polymer networks crosslinked by dynamic bonds

Dynamic polymer networks (DPNs) crosslinked by dynamic bonds have received intensive attention because of their special crack-healing capability. Diverse DPNs have been synthesized using a number of dynamic bonds, including dynamic covalent bond, hydrogen bond, ionic bond, metal-ligand coordination, hydrophobic interaction, and others. Despite the promising success in the polymer synthesis, the fundamental understanding of their self-healing mechanics is still at the very beginning. Especially, a general analytical model to understand the interfacial self-healing behaviors of DPNs has not been established. Here, we develop polymer-network based analytical theories that can mechanistically model the constitutive behaviors and interfacial self-healing behaviors of DPNs. We consider that the DPN is composed of interpenetrating networks crosslinked by dynamic bonds. The network chains follow inhomogeneous chain-length distributions and the dynamic bonds obey a force-dependent chemical kinetics. During the self-healing process, we consider the polymer chains diffuse across the interface to reform the dynamic bonds, being modeled by a diffusion-reaction theory. The theories can predict the stress-stretch behaviors of original and self-healed DPNs, as well as the healing strength in a function of healing time. We show that the theoretically predicted healing behaviors can consistently match the documented experimental results of DPNs with various dynamic bonds, including dynamic covalent bonds (diarylbibenzofuranone and olefin metathesis), hydrogen bonds, and ionic bonds. We expect our model to be a powerful tool for the self-healing community to invent, design, understand, and optimize self-healing DPNs with various dynamic bonds.