A general reaction–diffusion model of acidity in cancer invasion

We model the metabolism and behaviour of a developing cancer tumour in the context of its microenvironment, with the aim of elucidating the consequences of altered energy metabolism. Of particular interest is the Warburg Effect, a widespread preference in tumours for cytosolic glycolysis rather than oxidative phosphorylation for glucose breakdown, as yet incompletely understood. We examine a candidate explanation for the prevalence of the Warburg Effect in tumours, the acid-mediated invasion hypothesis, by generalising a canonical non-linear reaction–diffusion model of acid-mediated tumour invasion to consider additional biological features of potential importance. We apply both numerical methods and a non-standard asymptotic analysis in a travelling wave framework to obtain an explicit understanding of the range of tumour behaviours produced by the model and how fundamental parameters govern the speed and shape of invading tumour waves. Comparison with conclusions drawn under the original system—a special case of our generalised system—allows us to comment on the structural stability and predictive power of the modelling framework.     

A bioreaction–diffusion model for growth of marine sponge explants in bioreactors

Marine sponges are sources of high-value bioactives. Engineering aspects of in vitro culture of sponges from cuttings (explants) are poorly understood. This work develops a diffusion-controlled growth model for sponge explants. The model assumes that the explant growth is controlled by diffusive transport of at least some nutrients from the surrounding medium into the explant that generally has a poorly developed aquiferous system for internal irrigation during early stages of growth. Growth is assumed to obey Monod-type kinetics. The model is shown to satisfactorily explain the measured growth behavior of the marine sponge Crambe crambe in two different growth media. In addition, the model is generally consistent with published data for growth of explants of the sponges Disidea avara and Hemimycale columella. The model predicted that nutrient concentration profiles for nutrients, such as dissolved oxygen within the explant, are consistent with data published by independent researchers. In view of the proposed model’s ability to explain available data for growth of several species of sponge explants, diffusive transport does play a controlling role in explant growth at least until a fully developed aquiferous system has become established. According to the model and experimental observations, the instantaneous growth rate depends on the size of the explant and all those factors that influence the diffusion of critical nutrients within the explant. Growth follows a hyperbolic profile that is consistent with the Monod kinetics.     

Evolution of conditional dispersal: a reaction–diffusion–advection model

To study evolution of conditional dispersal, a Lotka-Volterra reaction-diffusion-advection model for two competing species in a heterogeneous environment is proposed and investigated. The two species are assumed to be identical except their dispersal strategies: both species disperse by random diffusion and advection along environmental gradients, but one species has stronger biased movement (i.e., advection along the environmental gradients) than the other one. It is shown that at least two scenarios can occur: if only one species has a strong tendency to move upward the environmental gradients, the two species can coexist since one species mainly pursues resources at places of locally most favorable environments while the other relies on resources from other parts of the habitat; if both species have such strong biased movements, it can lead to overcrowding of the whole population at places of locally most favorable environments, which causes the extinction of the species with stronger biased movement. These results provide a new mechanism for the coexistence of competing species, and they also imply that selection is against excessive advection along environmental gradients, and an intermediate biased movement rate may evolve.     

A multicomponent reaction–diffusion model of a heterogeneously distributed immobilized enzyme

A physical model was derived for the synthesis of the antibiotic cephalexin with an industrial immobilized penicillin G acylase, called Assemblase. In reactions catalyzed by Assemblase, less product and more by-product are formed in comparison with a free-enzyme catalyzed reaction. The model incorporates reaction with a heterogeneous enzyme distribution, electrostatically coupled transport, and pH-dependent dissociation behavior of reactants and is used to obtain insight in the complex interplay between these individual processes leading to the suboptimal conversion. The model was successfully validated with synthesis experiments for conditions ranging from heavily diffusion limited to hardly diffusion limited, including substrate concentrations from 50 to 600 mM, temperatures between 273 and 303 K, and pH values between 6 and 9. During the conversion of the substrates into cephalexin, severe pH gradients inside the biocatalytic particle, which were previously measured by others, were predicted. Physical insight in such intraparticle process dynamics may give important clues for future biocatalyst design. The modular construction of the model may also facilitate its use for other bioconversions with other biocatalysts.     

A reaction–diffusion malaria model with incubation period in the vector population

Malaria is one of the most important parasitic infections in humans and more than two billion people are at risk every year. To understand how the spatial heterogeneity and extrinsic incubation period (EIP) of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction–diffusion model. We then define the basic reproduction ratio R₀{\mathcal{R}_0} and show that R₀{\mathcal{R}_0} serves as a threshold parameter that predicts whether malaria will spread. Furthermore, a sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Numerically, we show that the use of the spatially averaged system may highly underestimate the malaria risk. The spatially heterogeneous framework in this paper can be used to design the spatial allocation of control resources.     

A new method for choosing the computational cell in stochastic reaction–diffusion systems

How to choose the computational compartment or cell size for the stochastic simulation of a reaction–diffusion system is still an open problem, and a number of criteria have been suggested. A generalized measure of the noise for finite-dimensional systems based on the largest eigenvalue of the covariance matrix of the number of molecules of all species has been suggested as a measure of the overall fluctuations in a multivariate system, and we apply it here to a discretized reaction–diffusion system. We show that for a broad class of first-order reaction networks this measure converges to the square root of the reciprocal of the smallest mean species number in a compartment at the steady state. We show that a suitably re-normalized measure stabilizes as the volume of a cell approaches zero, which leads to a criterion for the maximum volume of the compartments in a computational grid. We then derive a new criterion based on the sensitivity of the entire network, not just of the fastest step, that predicts a grid size that assures that the concentrations of all species converge to a spatially-uniform solution. This criterion applies for all orders of reactions and for reaction rate functions derived from singular perturbation or other reduction methods, and encompasses both diffusing and non-diffusing species. We show that this predicts the maximal allowable volume found in a linear problem, and we illustrate our results with an example motivated by anterior-posterior pattern formation in Drosophila, and with several other examples.     

Stationary multiple spots for reaction–diffusion systems

In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction–diffusion systems. We will consider two classes of reaction–diffusion systems: activator–inhibitor systems (such as the Gierer–Meinhardt system) and activator–substrate systems (such as the Gray–Scott system or the Schnakenberg model). The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature. We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator. Existence of multi-spots is proved using tools from nonlinear functional analysis such as Liapunov–Schmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis. Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots. Eigenvalues converging to zero are investigated using a projection similar to Liapunov–Schmidt reduction and conditions on the positions for stable spots are derived. The Green’s function of the Laplacian plays a central role in the analysis. The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations.      

Deriving reaction–diffusion models in ecology from interacting particle systems

We use a scaling procedure based on averaging Poisson distributed random variables to derive population level models from local models of interactions between individuals. The procedure is suggested by using the idea of hydrodynamic limits to derive reaction-diffusion models for population interactions from interacting particle systems. The scaling procedure is formal in the sense that we do not address the issue of proving that it converges; instead we focus on methods for computing the results of the scaling or deriving properties of rescaled systems. To that end we treat the scaling procedure as a transform, in analogy with the Laplace or Fourier transform, and derive operational formulas to aid in the computation of rescaled systems or the derivation of their properties. Since the limiting procedure is adapted from work by Durrett and Levin, we refer to the transform as the Durrett-Levin transform. We examine the effects of rescaling in various standard models, including Lotka-Volterra models, Holling type predator-prey models, and ratio-dependent models. The effects of scaling are mostly quantitative in models with smooth interaction terms, but ratio-dependent models are profoundly affected by the scaling. The scaling transforms ratio-dependent terms that are singular at the origin into smooth terms. Removing the singularity at the origin eliminates some of the unique dynamics that can arise in ratio-dependent models.Research partially supported by NSF grants DMS 99-73017 and DMS 02-11367     

A Lagrangian particle method for reaction–diffusion systems on deforming surfaces

Reaction–diffusion processes on complex deforming surfaces are fundamental to a number of biological processes ranging from embryonic development to cancer tumor growth and angiogenesis. The simulation of these processes using continuum reaction–diffusion models requires computational methods capable of accurately tracking the geometric deformations and discretizing on them the governing equations. We employ a Lagrangian level-set formulation to capture the deformation of the geometry and use an embedding formulation and an adaptive particle method to discretize both the level-set equations and the corresponding reaction–diffusion. We validate the proposed method and discuss its advantages and drawbacks through simulations of reaction–diffusion equations on complex and deforming geometries.     

Optimal control of vaccination in a vector-borne reaction–diffusion model applied to Zika virus

Journal of Mathematical Biology - Zika virus has acquired worldwide concern after a recent outbreak in Latin America that started in Brazil, with associated neurological conditions such as...     

Persistence and extinction of population in reaction–diffusion–advection model with strong Allee effect growth

Journal of Mathematical Biology - A reaction–diffusion–advection equation with strong Allee effect growth rate is proposed to model a single species stream population in a...     

Patterns of patchy spread in multi-species reaction–diffusion models

Spread of populations in space often takes place via formation, interaction and propagation of separated patches of high species density, without formation of continuous fronts. This type of spread is called a ‘patchy spread’. In earlier models, this phenomenon was considered to be a result of a pronounced environmental or/and demographic stochasticity. Recently, it was found that a patchy spread can arise in a fully deterministic predator–prey system and in models of infectious diseases; in each case the process takes place in a homogeneous environment. It is well recognized that the observed patterns of patchy spread in nature are a result of interplay between stochastic and deterministic factors. However, the models considering deterministic mechanism of patchy spread are developed and studied much less compared to those based on stochastic mechanisms. A further progress in the understanding of the role of deterministic factors in the patchy spread would be extremely helpful. Here we apply multi-species reaction–diffusion models of two spatial dimensions in a homogeneous environment. We demonstrate that patterns of patchy spread are rather common for the considered approach, in particular, they arise both in mutualism and competition models influenced by predation. We show that this phenomenon can occur in a system without a strong Allee effect, contrary to what was assumed to be crucial in earlier models. We show, as well, a pattern of patchy spread having significantly different speeds in different spatial directions. We analyze basic features of spatiotemporal dynamics of patchy spread common for the reaction–diffusion approach. We discuss in which ecosystems we would observe patterns of deterministic patchy spread due to the considered mechanism.     

A computational reaction–diffusion model for biosynthesis and linking of cartilage extracellular matrix in cell-seeded scaffolds with varying porosity

Biomechanics and Modeling in Mechanobiology - Cartilage tissue engineering is commonly initiated by seeding cells in porous materials such as hydrogels or scaffolds. Under optimal conditions, the...     

Wavespeed in reaction–diffusion systems, with applications to chemotaxis and population pressure

We present a method based on the Melnikov function used in dynamical systems theory to determine the wavespeed of travelling waves in perturbed reaction–diffusion systems. We study reaction–diffusion systems which are subject to weak nontrivial perturbations in the reaction kinetics, in the diffusion coefficient, or with weak active advection. We find explicit formulæ for the wavespeed and illustrate our theory with two examples; one in which chemotaxis gives rise to nonlinear advection and a second example in which a positive population pressure results in both a density-dependent diffusion coefficient and a nonlinear advection. Based on our theoretical results we suggest an experiment to distinguish between chemotactic and population pressure in bacterial colonies.     

Reaction–diffusion model of atherosclerosis development

Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines. The model represents a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction-diffusion wave. The theoretical results are confirmed by the results of numerical simulations.     

Global existence for semilinear reaction–diffusion systems on evolving domains

We present global existence results for solutions of reaction–diffusion systems on evolving domains. Global existence results for a class of reaction–diffusion systems on fixed domains are extended to the same systems posed on spatially linear isotropically evolving domains. The results hold without any assumptions on the sign of the growth rate. The analysis is valid for many systems that commonly arise in the theory of pattern formation. We present numerical results illustrating our theoretical findings.     

A comparison of the photosynthesis — light intensity relationship in phylogenetically different marine microalgae

Measured under equivalent physiological conditions, the photosynthesis-light intensity relationship based on oxygen production/mg chlorophyll a was found to be the same in five species representing the different chlorophyll c containing divisions of marine phytoplankton. Other non-photochemical metabolic processes related to photosynthesis such as diurnal variations, maximal photosynthesis rates, and dark oxygen uptake were quite different, and so these are the more significant factors in production and ecological distribution of diatoms, dinoflagellates, and coccolithophores. In contrast, the green algae tested showed a significantly different photosynthesis-light intensity curve from the chlorophyll c group.