Efficient and scalable prediction of stochastic reaction-diffusion processes using graph neural networks.

The dynamics of locally interacting particles that are distributed in space give rise to a multitude of complex behaviors. However the simulation of reaction-diffusion processes which model such systems is highly computationally expensive, the cost increasing rapidly with the size of space. Here, we devise a graph neural network based approach that uses cheap Monte Carlo simulations of reaction-diffusion processes in a small space to cast predictions of the dynamics of the same processes in a much larger and complex space, including spaces modelled by networks with heterogeneous topology. By applying the method to two biological examples, we show that it leads to accurate results in a small fraction of the computation time of standard stochastic simulation methods. The scalability and accuracy of the method suggest it is a promising approach for studying reaction-diffusion processes in complex spatial domains such as those modelling biochemical reactions, population evolution and epidemic spreading.

Sliding mode control for uncertain fractional-order reaction-diffusion memristor neural networks with time delays.

This paper investigates a sliding mode control method for a class of uncertain delayed fractional-order reaction-diffusion memristor neural networks. Different from most existing literature on sliding mode control for fractional-order reaction-diffusion systems, this study constructs a linear sliding mode switching function and designs the corresponding sliding mode control law. The sufficient theory for the globally asymptotic stability of the sliding mode dynamics are provided, and it is proven that the sliding mode surface is finite-time reachable under the proposed control law, with an estimate of the maximum reaching time. Finally, a numerical test is presented to validate the effectiveness of the theoretical analysis.

Entropy Production in Reaction-Diffusion Systems Confined in Narrow Channels.

This work analyzes the effect of wall geometry when a reaction-diffusion system is confined to a narrow channel. In particular, we study the entropy production density in the reversible Gray-Scott system. Using an effective diffusion equation that considers modifications by the channel characteristics, we find that the entropy density changes its value but not its qualitative behavior, which helps explore the structure-formation space.

A mathematical model of signalling molecule-mediated processes during regeneration of osteochondral defects after chondrocyte implantation.

Treating bone-cartilage defects is a fundamental clinical problem. The ability of damaged cartilage to self-repair is limited due to its avascularity. Left untreated, these defects can lead to osteoarthritis. Details of osteochondral defect repair are elusive, but animal models indicate healing occurs via an endochondral ossification-like process, similar to that in the growth plate. In the growth plate, the signalling molecules parathyroid hormone-related protein (PTHrP) and Indian Hedgehog (Ihh) form a feedback loop regulating chondrocyte hypertrophy, with Ihh inducing and PTHrP suppressing hypertrophy. To better understand this repair process and to explore the regulatory role of signalling molecules on the regeneration process, we formulate a reaction-diffusion mathematical model of osteochondral defect regeneration after chondrocyte implantation. The drivers of healing are assumed to be chondrocytes and osteoblasts, and their interaction via signalling molecules. We model cell proliferation, migration and chondrocyte hypertrophy, and matrix production and conversion, spatially and temporally. We further model nutrient and signalling molecule diffusion and their interaction with the cells. We consider the PTHrP-Ihh feedback loop as the backbone mechanisms but the model is flexible to incorporate extra signalling mechanisms if needed. Our mathematical model is able to represent repair of osteochondral defects, starting with cartilage formation throughout the defect. This is followed by chondrocyte hypertrophy, matrix calcification and bone formation deep inside the defect, while cartilage at the surface is maintained and eventually separated from the deeper bone by a thin layer of calcified cartilage. The complete process requires around 48 months. A key highlight of the model demonstrates that the PTHrP-Ihh loop alone is insufficient and an extra mechanism is required to initiate chondrocyte hypertrophy, represented by a critical cartilage density. A parameter sensitivity study reveals that the timing of the repair process crucially depends on parameters, such as the critical cartilage density, and those describing the actions of PTHrP to suppress hypertrophy, such as its diffusion coefficient, threshold concentration and degradation rate.

Phytoplankton-chytrid-zooplankton dynamics via a reaction-diffusion-advection mycoloop model.

Mycoloop is an important aquatic food web composed of phytoplankton, chytrids (one dominant group of parasites in aquatic ecosystems), and zooplankton. Chytrids infect phytoplankton and fragment them for easy consumption by zooplankton. The free-living chytrid zoospores are also a food resource for zooplankton. A dynamic reaction-diffusion-advection mycoloop model is proposed to describe the Phytoplankton-chytrid-zooplankton interactions in a poorly mixed aquatic environment. We analyze the dynamics of the mycoloop model to obtain dissipativity, steady state solutions, and persistence. We rigorously derive several critical thresholds for phytoplankton or zooplankton invasion and chytrid transmission among phytoplankton. Numerical diagrams show that varying ecological factors affect the formation and breakup of the mycoloop, and zooplankton can inhibit chytrid transmission among phytoplankton. Furthermore, this study suggests that mycoloop may either control or cause phytoplankton blooms.

Performance characteristics allow for confinement of a CRISPR toxin-antidote gene drive for population suppression in a reaction-diffusion model.

Gene drive alleles that can bias their own inheritance could engineer populations for control of disease vectors, invasive species and agricultural pests. There are successful examples of suppression drives and confined modification drives, but developing confined suppression drives has proven more difficult. However, CRISPR-based toxin-antidote dominant embryo (TADE) suppression drive may fill this niche. It works by targeting and disrupting a haplolethal target gene in the germline with its gRNAs while rescuing this target. It also disrupts a female fertility gene by driving insertion or additional gRNAs. Here, we used a reaction-diffusion model to assess drive performance in continuous space, where outcomes can be substantially different from those in panmictic populations. We measured drive wave speed and found that moderate fitness costs or target gene disruption in the early embryo from maternally deposited nuclease can eliminate the drive's ability to form a wave of advance. We assessed the required release size, and finally we investigated migration corridor scenarios. It is often possible for the drive to suppress one population and then persist in the corridor without invading the second population, a potentially desirable outcome. Thus, even imperfect variants of TADE suppression drive may be excellent candidates for confined population suppression.

Oscillations in a Spatial Oncolytic Virus Model.

Virotherapy treatment is a new and promising target therapy that selectively attacks cancer cells without harming normal cells. Mathematical models of oncolytic viruses have shown predator-prey like oscillatory patterns as result of an underlying Hopf bifurcation. In a spatial context, these oscillations can lead to different spatio-temporal phenomena such as hollow-ring patterns, target patterns, and dispersed patterns. In this paper we continue the systematic analysis of these spatial oscillations and discuss their relevance in the clinical context. We consider a bifurcation analysis of a spatially explicit reaction-diffusion model to find the above mentioned spatio-temporal virus infection patterns. The desired pattern for tumor eradication is the hollow ring pattern and we find exact conditions for its occurrence. Moreover, we derive the minimal speed of travelling invasion waves for the cancer and for the oncolytic virus. Our numerical simulations in 2-D reveal complex spatial interactions of the virus infection and a new phenomenon of a periodic peak splitting. An effect that we cannot explain with our current methods.

Modeling the reactive oxygen species (ROS) wave in Chlamydomonas reinhardtii colonies.

Reactive oxygen species (ROS) play a crucial role as signaling molecules in both plant and animal cells, enabling rapid responses to various stimuli. Among the many cellular mechanisms used to generate and transduce ROS signals, ROS-induced-ROS release (RIRR) is emerging as an important pathway involved in the responses of various multicellular and unicellular organisms to environmental stresses. In RIRR, one cellular compartment, organelle, or cell generates or releases ROS, triggering an increased ROS production and release by another compartment, organelle, or cell, thereby giving rise to a fast propagating ROS wave. This RIRR-based signal relay has been demonstrated to facilitate mitochondria-to-mitochondria communication in animal cells and long-distance systemic signaling in plants in response to biotic and abiotic stresses. More recently, it has been discovered that different unicellular microorganism communities also exhibit a RIRR cell-to-cell signaling process triggered by a localized stress treatment. However, the precise mechanism underlying the propagation of the ROS signal among cells within these unicellular communities remained elusive. In this study, we employed a reaction-diffusion model incorporating the RIRR mechanism to analyze the propagation of ROS-mediated signals. By effectively balancing production and scavenging processes, our model successfully reproduces the experimental ROS signal velocities observed in unicellular green algae (Chlamydomonas reinhardtii) colonies grown on agar plates, furthering our understanding of intercellular ROS communication.

Preparation of surgical meshes using self-regulating technology based on reaction-diffusion processes.

While reaction-diffusion processes are utilized in multiple scientific fields, these phenomena have seen limited practical application in the polymer industry. Although self-regulating processes driven by parallel reaction and diffusion can lead to patterned structures, most polymeric products with repeating subunits are still prepared by methods that require complex and expensive instrumentation. A notable, high-added-value example is surgical mesh, which is often manufactured by weaving or knitting. In our present work, we demonstrate how the polymer and the biomedical industry can benefit from the pattern-forming capabilities of reaction-diffusion. We would like to propose a self-regulating method that facilitates the creation of surgical meshes from biocompatible polymers. Since the control of the process assumes a thorough understanding of the underlying phenomena, the theoretical background, as well as a mathematical model that can accurately describe the empirical data, is also introduced and explained. Our method offers the benefits of conventional techniques while introducing additional advantages not attainable with them. Most importantly, the method proposed in this paper enables the rapid creation of meshes with an average pore size that can be adjusted easily and tailored to fit the intended area of application.

Reaction-diffusion study of electron-beam-induced contamination growth.

A time-dependent reaction-diffusion model was elaborated to better understand the dynamical growth of contamination on surfaces illuminated by an electron beam. The goal of this work was to fully describe the flow of hydrocarbon molecules, denoted as contaminants, and their polymerization in the irradiated area with the number of parameters reduced to a minimum necessary. It was considered that the diffusion process of contaminants is driven by the gradient of their surface density generated by the impact of a circular homogeneous electron beam. The contribution of the residual gas atmosphere in the instrument was described by the tendency to reestablish the initial equilibrium surface density of contaminants before irradiation. The four unknown parameters of the model, the electron interaction cross-section, the diffusion coefficient, the initial surface density of contaminants, and the frequency of the supply of contaminants from the residual gas atmosphere were determined by comparing the modeled contamination growth with experimental results. The experiments were designed such that the influence of the single parameters could be unequivocally separated. To follow the dynamical evolution of the system and to generate time-resolved distinct experimental data, successive contamination measurements were performed at short time intervals up to 20 min. The local height and shape of the grown contamination were quantified by evaluating high-angle annular dark-field (HAADF) scanning-transmission- electron-microcopy (STEM) image intensities and corresponding Monte-Carlo simulations. Our model also applies to nonhomogeneous initial conditions like the reduced local surface density of contaminants after previous beam-showering. The dynamic analyses of this process might provide hints regarding the relative size of the contaminant molecules and also indicate some measures for the reduction of contamination growth.

Emergence of non-trivial solutions from trivial solutions in reaction-diffusion equations for pattern formation.

Reaction-diffusion equations serve as fundamental tools in describing pattern formation in biology. In these models, nonuniform steady states often represent stationary spatial patterns. Notably, these steady states are not unique, and unveiling them mathematically presents challenges. In this paper, we introduce a framework based on bifurcation theory to address pattern formation problems, specifically examining whether nonuniform steady states can bifurcate from trivial ones. Furthermore, we employ linear stability analysis to investigate the stability of the trivial steady-state solutions. We apply the method to two classic reaction-diffusion models: the Schnakenberg model and the Gray-Scott model. For both models, our approach effectively reveals many nonuniform steady states and assesses the stability of the trivial solution. Numerical computations are also presented to validate the solution structures for these models.

Chemical reaction steers spatiotemporal self-assembly of supramolecular hydrogels.

Supramolecular structures are widespread in living system, which are usually spatiotemporally regulated by sophisticated metabolic processes to enable vital biological functions. Inspired by living system, tremendous efforts have been made to realize spatiotemporal control over the self-assembly of supramolecular materials in synthetic scenario by coupling chemical reaction with molecular self-assembly process. In this review, we focused on the works related to supramolecular hydrogels that are regulated in space and time using chemical reaction. Firstly, we summarized how spatially controlled self-assembly of supramolecular hydrogels can be achieved via chemical reaction-instructed self-assembly, and the application of such a self-assembly methodology in biotherapy was discussed as well. Second, we reviewed dynamic supramolecular hydrogels dictated by chemical reaction networks that can evolve their structures and properties against time. Third, we discussed the recent progresses in the control of the self-assembly of supramolecular hydrogels in both space and time though a reaction-diffusion-coupled self-assembly approach. Finally, we provided a perspective on the further development of spatiotemporally controlled supramolecular hydrogels using chemical reaction in the future.

Deciphering immunodiffusion: In silico optimization for faster protein diagnostics.

Immunodiffusion tests offer a simple yet powerful method for detecting protein antigens, but their long assay times hinder clinical utility. We unveil the complex interplay of parameters governing this process using finite element simulations. By meticulously validating our model against real-world data, we elucidate how initial concentrations and diffusivities of antigen and antibody shape the intensity, size, and formation time of the precipitin ring. Our key innovation lies in employing phase diagram analysis to map the combined effects of these parameters on assay performance. This framework enables rapid in silico parameter estimation, paving the way for the design of novel immunodiffusion assays with drastically reduced assay times. The presented approach holds immense potential for optimizing protein diagnostics for fast and reliable diagnostics.

Global analysis of a diffusive Cholera model with multiple transmission pathways, general incidence and incomplete immunity in a heterogeneous environment.

With the consideration of the complexity of the transmission of Cholera, a partially degenerated reaction-diffusion model with multiple transmission pathways, incorporating the spatial heterogeneity, general incidence, incomplete immunity, and Holling type Ⅱ treatment was proposed. First, the existence, boundedness, uniqueness, and global attractiveness of solutions for this model were investigated. Second, one obtained the threshold condition $ \mathcal{R}_{0} $ and gave its expression, which described global asymptotic stability of disease-free steady state when $ \mathcal{R}_{0} < 1 $, as well as the maximum treatment rate as zero. Further, we obtained the disease was uniformly persistent when $ \mathcal{R}_{0} > 1 $. Moreover, one used the mortality due to disease as a branching parameter for the steady state, and the results showed that the model undergoes a forward bifurcation at $ \mathcal{R}_{0} $ and completely excludes the presence of endemic steady state when $ \mathcal{R}_{0} < 1 $. Finally, the theoretical results were explained through examples of numerical simulations.

Local Self-Assembly of Dissipative Structures Sustained by Substrate Diffusion.

The coupling between energy-consuming molecular processes and the macroscopic dimension plays an important role in nature and in the development of active matter. Here, we study the temporal evolution of a macroscopic system upon the local activation of a dissipative self-assembly process. Injection of surfactant molecules in a substrate-containing hydrogel results in the local substrate-templated formation of assemblies, which are catalysts for the conversion of substrate into waste. We show that the system develops into a macroscopic (pseudo-)non-equilibrium steady state (NESS) characterized by the local presence of energy-dissipating assemblies and persistent substrate and waste concentration gradients. For elevated substrate concentrations, this state can be maintained for more than 4 days. The studies reveal an interdependence between the dissipative assemblies and the concentration gradients: catalytic activity by the assemblies results in sustained concentration gradients and, vice versa, continuous diffusion of substrate to the assemblies stabilizes their size. The possibility to activate dissipative processes with spatial control and create long lasting non-equilibrium steady states enables dissipative structures to be studied in the space-time domain, which is of relevance for understanding biological systems and for the development of active matter.

Cross Talking Calcium, IP3 and Buffer Dynamics Alters ATP and NADH Level in Obese and Normal Hepatocyte Cell.

The cross talk between calcium (Ca2+), IP3 and buffer dynamics regulate various mechanisms in hepatocyte cells. The study of independent systems of calcium, IP3, and buffer signaling provides limited information about cell dynamics. In the current study, coupled reaction-diffusion equations are used to design a cross-talk model for IP3, buffer, and calcium dynamics in a hepatocyte cell. The one-way feedback of calcium, buffer, and IP3 in ATP production, ATP degradation, and NADH production rate is incorporated into the model. Numerical simulation has been done using the Finite Element Method (FEM) along the spatial direction and the Crank-Nicolson (C-N) method along the temporal direction. The numerical results are analysed to determine the effects of alterations in processes of cross-talking dynamics of IP3, buffer, and calcium on ATP and NADH production and degradation rate of ATP in a hepatocyte cell under normal and obesity conditions. The comparative analysis of these findings unveils notable distinctions induced by obesity in calcium dynamics, ATP and NADH synthesis, and ATP degradation kinetics.

A population model-based linear-quadratic Gaussian compensator for the control of intravenously infused alcohol studies and withdrawal symptom prophylaxis using transdermal sensing.

An output feedback LQG compensator (combined controller and state estimator) for the regulation of intravenous-infused alcohol studies and treatment using a noninvasive transdermal alcohol biosensor is developed. The design is based on a population model involving an abstract semi-linear parabolic hybrid reaction-diffusion system involving coupled partial and ordinary differential equations with random parameters known only up to their distributions. The scheme developed is based on a weak formulation of the model equations in an appropriately constructed Gelfand triple of Bochner spaces wherein the unknown random parameters are treated as additional spatial variables. Implementation relies on a Galerkin-based approximation and convergence theory and an abstract formulation involving linear semigroups of operators. The model is fit and validated using laboratory collected human subject data and the method of moments. The results of numerical simulations of controlled intravenous alcohol infusion are presented and discussed.

Modelling phytoplankton-virus interactions: phytoplankton blooms and lytic virus transmission.

A dynamic reaction-diffusion model of four variables is proposed to describe the spread of lytic viruses among phytoplankton in a poorly mixed aquatic environment. The basic ecological reproductive index for phytoplankton invasion and the basic reproduction number for virus transmission are derived to characterize the phytoplankton growth and virus transmission dynamics. The theoretical and numerical results from the model show that the spread of lytic viruses effectively controls phytoplankton blooms. This validates the observations and experimental results of Emiliana huxleyi-lytic virus interactions. The studies also indicate that the lytic virus transmission cannot occur in a low-light or oligotrophic aquatic environment.

Travelling waves for a fast reaction limit of a discrete coagulation-fragmentation model with diffusion and proliferation.

We study traveling wave solutions for a reaction-diffusion model, introduced in the article Calvez et al. (Regime switching on the propagation speed of travelling waves of some size-structured myxobacteriapopulation models, 2023), describing the spread of the social bacterium Myxococcus xanthus. This model describes the spatial dynamics of two different cluster sizes: isolated bacteria and paired bacteria. Two isolated bacteria can coagulate to form a cluster of two bacteria and conversely, a pair of bacteria can fragment into two isolated bacteria. Coagulation and fragmentation are assumed to occur at a certain rate denoted by k. In this article we study theoretically the limit of fast coagulation fragmentation corresponding mathematically to the limit when the value of the parameter k tends to + ∞ . For this regime, we demonstrate the existence and uniqueness of a transition between pulled and pushed fronts for a certain critical ratio θ ⋆ between the diffusion coefficient of isolated bacteria and the diffusion coefficient of paired bacteria. When the ratio is below θ ⋆ , the critical front speed is constant and corresponds to the linear speed. Conversely, when the ratio is above the critical threshold, the critical spreading speed becomes strictly greater than the linear speed.

How do productivity gradient and diffusion shape patterns in a plant-herbivore grazing system?

Natural systems show heterogeneous patchy distributions of vegetation over large landscapes. Reaction-diffusion systems can demonstrate such heterogeneity of species distributions. Here, we analyse a reaction-diffusion model of plant-herbivore interactions in two-dimensional space to illustrate non-homogeneous distributions of plants and herbivores. The non-spatial system shows bottom-up control, where herbivore density is low under low and high primary productivity but increased at intermediate productivity. In addition, the non-spatial system provides bistability between a dense vegetation state devoid of herbivores and a coexisting state of plants and herbivores. In the spatiotemporal model, we give analytical conditions of occurring diffusion-driven (Turing) instability, where a novel point in our model is the relative dispersal of herbivores, which represents the movement of herbivores from a higher to a lower vegetation state in addition to the self-diffusion of both species. It is shown that heterogeneity in the population distribution does not occur if the relative dispersal of herbivores is low, but it appears in the opposite case. Due to bistability in the underlying non-spatial system, the spatiotemporal model produces initial value-dependent patterns. The two initial values make different patterns despite having the same primary productivity and relative dispersal rate. As productivity increases with a given relative herbivore dispersal, pattern transition occurs from a blend of stripes and spots of low vegetation state to a predominantly low-density vegetation state with smaller patches of densely vegetated states with one initial value. On the contrary, a discernible change in vegetation patterns from cold spots in the dense vegetation to hot stripes in the primarily low-vegetated state is noticed under the other initial population value. Furthermore, the population distributions of plants and herbivores in the entire domain after a long period are heterogeneous for both initial values, provided the relative herbivore dispersal is substantial. We estimated mean population densities to observe species fitness in the whole domain under variable productivity. When productivity is high, the mean population density of plants may go up or down, depending on the herbivore's relative dispersal rate. In contrast to the bottom-up control dynamics of the non-spatial system, the system exhibits a top-down control under high relative dispersal, where the herbivore regulates vegetation growth under high productivity. On the other hand, herbivores are extinct under high productivity if the relative dispersal is low.