Reaction–diffusion models of development with state-dependent chemical diffusion coefficients

Quantitative and analytical tools to analyze the spatiotemporal population dynamics of microbial consortia

Microorganisms occupy almost every niche on earth. They play critical roles in maintaining ecological balance, atmospheric C/N cycle, and human health. Microbes live in consortia with metabolite exchange or signal communication. Quantitative and analytical tools are becoming increasingly important to study microbial consortia dynamics. We argue that a combined reductionist and holistic approach will be important to understanding the assembly rules and spatiotemporal population dynamics of the microbial community (MICOM). Reductionism allows us to reconstruct complex MICOM from isolated or simple synthetic consortia. Holism allows us to understand microbes as a community with cooperation and competition. Here we review the recent development of quantitative and analytical tools to uncover the underlying principles in microbial communities that govern their spatiotemporal change and interaction dynamics. Mathematical models and analytical tools will continue to provide essential knowledge and expand our capability to manipulate and control microbial consortia.

Cellular Automata Modeling of Stem-Cell-Driven Development of Tissue in the Nervous System

Mathematical and computational modeling enables biologists to integrate data from observations and experiments into a theoretical framework. In this review, we describe how developmental processes associated with stem-cell-driven growth of tissue in both the embryonic and adult nervous system can be modeled using cellular automata (CA). A cellular automaton is defined by its discrete nature in time, space, and state. The discrete space is represented by a uniform grid or lattice containing agents that interact with other agents within their local neighborhood. This possibility of local interactions of agents makes the cellular automata approach particularly well suited for studying through modeling how complex patterns at the tissue level emerge from fundamental developmental processes (such as proliferation, migration, differentiation, and death) at the single-cell level. As part of this review, we provide a primer for how to define biologically inspired rules governing these processes so that they can be implemented into a CA model. We then demonstrate the power of the CA approach by presenting simulations (in the form of figures and movies) based on building models of three developmental systems: the formation of the enteric nervous system through invasion by neural crest cells; the growth of normal and tumorous neurospheres induced by proliferation of adult neural stem/progenitor cells; and the neural fate specification through lateral inhibition of embryonic stem cells in the neurogenic region of Drosophila.

Stability and convergence of the variable directions difference scheme for one nonlinear two-dimensional model

The system of two-dimensional nonlinear partial differential equations is considered. This system describes the vein formation in meristematic tissues of young leaves. Variable directions difference scheme is constructed and investigated. Absolute stability regarding space and time steps of scheme is shown. The convergence statement for the constructed scheme is proved. Rate of convergence is given. Various numerical experiments are carried out and results of some of them are considered in this paper. Comparison of numerical experiments with the results of the theoretical investigation is given too.

Turing pattern formation in the Brusselator system with nonlinear diffusion

In this work we investigate the effect of density-dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern formation. We study the process of pattern formation both in one-dimensional and two-dimensional spatial domains. Through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations shows the occurrence of a number of different phenomena, including stable supercritical and subcritical Turing patterns with multiple branches of stable solutions leading to hysteresis. Moreover, we consider traveling patterning waves: When the domain size is large, the pattern forms sequentially and traveling wave fronts are the precursors to patterning. We derive the Ginzburg-Landau equation and describe the traveling front enveloping a pattern which invades the domain. We show the emergence of radially symmetric target patterns, and, through a matching procedure, we construct the outer amplitude equation and the inner core solution.

Simulation of mRNA diffusion in the nuclear environment

A mathematical model is devised to study the diffusion of mRNA in the nucleus from the site of synthesis to a nuclear pore where it is exported to the cytoplasm. This study examines the role that nuclear structure can play in determining the kinetics of export by considering models in which elements of the nuclear skeleton and confinement by chromatin direct the mRNA movement. As a rule, a dense chromatin layer favours rapid export by reducing the effective volume for diffusion. However, it may also result in a heavy tail in the export time distribution because of the low mobility of molecules that accidentally find their way deep into the dense layer. An anisotropic solid-state transport system can also assist export. There exist both an optimal ratio of the anisotropy and an optimal depth of the solid-state transport layer that favour rapid export.

Modelling non-homogeneous stochastic reaction-diffusion systems: the case study of gemcitabine-treated non-small cell lung cancer growth

Background

Reaction-diffusion based models have been widely used in the literature for modeling the growth of solid tumors. Many of the current models treat both diffusion/consumption of nutrients and cell proliferation. The majority of these models use classical transport/mass conservation equations for describing the distribution of molecular species in tumor spheroids, and the Fick's law for describing the flux of uncharged molecules (i.e oxygen, glucose). Commonly, the equations for the cell movement and proliferation are first order differential equations describing the rate of change of the velocity of the cells with respect to the spatial coordinates as a function of the nutrient's gradient. Several modifications of these equations have been developed in the last decade to explicitly indicate that the tumor includes cells, interstitial fluids and extracellular matrix: these variants provided a model of tumor as a multiphase material with these as the different phases. Most of the current reaction-diffusion tumor models are deterministic and do not model the diffusion as a local state-dependent process in a non-homogeneous medium at the micro- and meso-scale of the intra- and inter-cellular processes, respectively. Furthermore, a stochastic reaction-diffusion model in which diffusive transport of the molecular species of nutrients and chemotherapy drugs as well as the interactions of the tumor cells with these species is a novel approach. The application of this approach to he scase of non-small cell lung cancer treated with gemcitabine is also novel.

Methods

We present a stochastic reaction-diffusion model of non-small cell lung cancer growth in the specification formalism of the tool Redi, we recently developed for simulating reaction-diffusion systems. We also describe how a spatial gradient of nutrients and oncological drugs affects the tumor progression. Our model is based on a generalization of the Fick's first diffusion law that allows to model diffusive transport in non-homogeneous media. The diffusion coefficient is explicitly expressed as a function depending on the local conditions of the medium, such as the concentration of molecular species, the viscosity of the medium and the temperature. We incorporated this generalized law in a reaction-based stochastic simulation framework implementing an efficient version of Gillespie algorithm for modeling the dynamics of the interactions between tumor cell, nutrients and gemcitabine in a spatial domain expressing a nutrient and drug concentration gradient.

Results

Using the mathematical framework of model we simulated the spatial growth of a 2D spheroidal tumor model in response to a treatment with gemcitabine and a dynamic gradient of oxygen and glucose. The parameters of the model have been taken from recet literature and also inferred from real tumor shrinkage curves measured in patients suffering from non-small cell lung cancer. The simulations qualitatively reproduce the time evolution of the morphologies of these tumors as well as the morphological patterns follow the growth curves observed in patients.

Conclusions

s This model is able to reproduce the observed increment/decrement of tumor size in response to the pharmacological treatment with gemcitabine. The formal specification of the model in Redi can be easily extended in an incremental way to include other relevant biophysical processes, such as local extracellular matrix remodelling, active cell migration and traction, and reshaping of host tissue vasculature, in order to be even more relevant to support the experimental investigation of cancer.

Regulation of Turing patterns in a spatially extended chlorine-iodine-malonic-acid system with a local concentration-dependent diffusivity

A chlorine-iodine-malonic-acid Turing system involving a local concentration-dependent diffusivity (LCDD) has fundamental significance for physical, chemical, and biological systems with inhomogeneous medium. We investigated such a system by both numerical computation and mathematical analysis. Our research reveals that a variable local diffusivity has an evident effect on regulating the Turing patterns for different modes. An intrinsic square-root law is given by λ ∼ (c(1)+c(2)k)(1/2), which relates the pattern wavelength (λ) with the LCDD coefficient (k). This law indicates that the system pattern has the properties of an equivalent Turing pattern. The current study confirms that, for the Turing system with LCDD, the system pattern form retains the basic characteristics of a traditional Turing pattern in a wide range of LCDD coefficients.

SELF-ASSEMBLED SCAFFOLDS USING REACTION–DIFFUSION SYSTEMS: A HYPOTHESIS FOR BONE REGENERATION

One area of tissue engineering concerns research into alternatives for new bone formation and replacing its function. Scaffolds have been developed to meet this requirement, allowing cell migration, bone tissue growth, transport of growth factors and nutrients, and the improvement of the mechanical properties of bone. Scaffolds are made from different biomaterials and manufactured using several techniques that, in some cases, do not allow full control over the size and orientation of the pores characterizing the scaffold. A novel hypothesis that a reaction–diffusion (RD) system can be used for designing the geometrical specifications of the bone matrix is thus presented here. The hypothesis was evaluated by making simulations in two- and three-dimensional RD systems in conjunction with the biomaterial scaffold. The results showed the methodology's effectiveness in controlling features such as the percentage of porosity, size, orientation, and interconnectivity of pores in an injectable bone matrix produced by the proposed hypothesis.

Continuum limits of pattern formation in hexagonal-cell monolayers

Intercellular signalling is key in determining cell fate. In closely packed tissues such as epithelia, juxtacrine signalling is thought to be a mechanism for the generation of fine-grained spatial patterns in cell differentiation commonly observed in early development. Theoretical studies of such signalling processes have shown that negative feedback between receptor activation and ligand production is a robust mechanism for fine-grained pattern generation and that cell shape is an important factor in the resulting pattern type. It has previously been assumed that such patterns can be analysed only with discrete models since significant variation occurs over a lengthscale concomitant with an individual cell; however, considering a generic juxtacrine signalling model in square cells, in O'Dea and King (Math Biosci 231(2):172-185 2011), a systematic method for the derivation of a continuum model capturing such phenomena due to variations in a model parameter associated with signalling feedback strength was presented. Here, we extend this work to derive continuum models of the more complex fine-grained patterning in hexagonal cells, constructing individual models for the generation of patterns from the homogeneous state and for the transition between patterning modes. In addition, by considering patterning behaviour under the influence of simultaneous variation of feedback parameters, we construct a more general continuum representation, capturing the emergence of the patterning bifurcation structure. Comparison with the steady-state and dynamic behaviour of the underlying discrete system is made; in particular, we consider pattern-generating travelling waves and the competition between various stable patterning modes, through which we highlight an important deficiency in the ability of continuum representations to accommodate certain dynamics associated with discrete systems.

The role of chemical dynamics in plant morphogenesis(1)

In biological development, the generation of shape is preceded by the spatial localization of growth factors. Localization, and how it is maintained or changed during the process of growth, determines the shapes produced. Mathematical models have been developed to investigate the chemical, mechanical and transport properties involved in plant morphogenesis. These synthesize biochemical and biophysical data, revealing underlying principles, especially the importance of dynamics in generating form. Chemical kinetics has been used to understand the constraints on reaction and transport rates to produce localized concentration patterns. This approach is well developed for understanding de novo pattern formation, pattern spacing and transitions from one pattern to another. For plants, growth is continual, and a key use of the theory is in understanding the feedback between patterning and growth, especially for morphogenetic events which break symmetry, such as tip branching. Within the context of morphogenetic modelling in general, the present review gives a brief history of chemical patterning research and its particular application to shape generation in plant development.

Mechanogenetic coupling of Hydra symmetry breaking and driven Turing instability model

The freshwater polyp Hydra can regenerate from tissue fragments or random cell aggregates. We show that the axis-defining step ("symmetry breaking") of regeneration requires mechanical inflation-collapse oscillations of the initial cell ball. We present experimental evidence that axis definition is retarded if these oscillations are slowed down mechanically. When biochemical signaling related to axis formation is perturbed, the oscillation phase is extended and axis formation is retarded as well. We suggest that mechanical oscillations play a triggering role in axis definition. We extend earlier reaction-diffusion models for Hydra regrowth by coupling morphogen transport to mechanical stress caused by the oscillations. The modified reaction-diffusion model reproduces well two important experimental observations: 1), the existence of an optimum size for regeneration, and 2), the dependence of the symmetry breaking time on the properties of the mechanical oscillations.

Are green islands red herrings? Significance of green islands in plant interactions with pathogens and pests

The term green island was first used to describe an area of living, green tissue surrounding a site of infection by an obligately biotrophic fungal pathogen, differentiated from neighbouring yellowing, senescent tissue. However, it has now been used to describe symptoms formed in response to necrotrophic fungal pathogens, virus infection and infestation by certain insects. In leaves infected by obligate biotrophs such as rust and powdery mildew pathogens, green islands are areas where senescence is retarded, photosynthetic activity is maintained and polyamines accumulate. We propose such areas, in which both host and pathogen cells are alive, be termed green bionissia. By contrast, we propose that green areas associated with leaf damage caused by toxins produced by necrotrophic fungal pathogens be termed green necronissia. A range of biotrophic/hemibiotrophic fungi and leaf-mining insects produce cytokinins and it has been suggested that this cytokinin secretion may be responsible for the green island formation. Indeed, localised cytokinin accumulation may be a common mechanism responsible for green island formation in interactions of plants with biotrophic fungi, viruses and insects. Models have been developed to study if green island formation is pathogen-mediated or host-mediated. They suggest that green bionissia on leaves infected by biotrophic fungal pathogens represent zones of host tissue, altered physiologically to allow the pathogen maximum access to nutrients early in the interaction, thus supporting early sporulation and increasing pathogen fitness. They lead to the suggestion that green islands are 'red herrings', representing no more than the consequence of the infection process and discrete changes in leaf senescence.

Toward realistic modeling of dynamic processes in cell signaling: quantification of macromolecular crowding effects

One of the major factors distinguishing molecular processes in vivo from biochemical experiments in vitro is the effect of the environment produced by macromolecular crowding in the cell. To achieve a realistic modeling of processes in the living cell based on biochemical data, it becomes necessary, therefore, to consider such effects. We describe a protocol based on Brownian dynamics simulation to characterize and quantify the effect of various forms of crowding on diffusion and bimolecular association in a simple model of interacting hard spheres. We show that by combining the elastic collision method for hard spheres and the mean field approach for hydrodynamic interaction (HI), our simulations capture the correct dynamics of a monodisperse system. The contributions from excluded volume effect and HI to the crowding effect are thus quantified. The dependence of the results on size distribution of each component in the system is illustrated, and the approach is applied as well to the crowding effect on electrostatic-driven association in both neutral and charged environments; values for effective diffusion constants and association rates are obtained for the specific conditions. The results from our simulation approach can be used to improve the modeling of cell signaling processes without additional computational burdens.

A multiscale theoretical model for diffusive mass transfer in cellular biological media

An integrated methodology is developed for the theoretical analysis of solute transport and reaction in cellular biological media, such as tissues, microbial flocs, and biofilms. First, the method of local spatial averaging with a weight function is used to establish the equation which describes solute conservation at the cellular biological medium scale, starting with a continuum-based formulation of solute transport at finer spatial scales. Second, an effective-medium model is developed for the self-consistent calculation of the local diffusion coefficient in the cellular biological medium, including the effects of the structural heterogeneity of the extra-cellular space and the reversible adsorption to extra-cellular polymers. The final expression for the local effective diffusion coefficient is: D(Abeta)=lambda(beta)D(Aupsilon), where D(Aupsilon) is the diffusion coefficient in water, and lambda(beta) is a function of the composition and fundamental geometric and physicochemical system properties, including the size of solute molecules, the size of extra-cellular polymer fibers, and the mass permeability of the cell membrane. Furthermore, the analysis sheds some light on the function of the extra-cellular hydrogel as a diffusive barrier to solute molecules approaching the cell membrane, and its implications on the transport of chemotherapeutic agents within a cellular biological medium. Finally, the model predicts the qualitative trend as well as the quantitative variability of a large number of published experimental data on the diffusion coefficient of oxygen in cell-entrapping gels, microbial flocs, biofilms, and mammalian tissues.

Approximate analytical time-dependent solutions to describe large-amplitude local calcium transients in the presence of buffers

Local Ca(2+) signaling controls many neuronal functions, which is often achieved through spatial localization of Ca(2+) signals. These nanodomains are formed due to combined effects of Ca(2+) diffusion and binding to the cytoplasmic buffers. In this article we derived simple analytical expressions to describe Ca(2+) diffusion in the presence of mobile and immobile buffers. A nonlinear character of the reaction-diffusion problem was circumvented by introducing a logarithmic approximation of the concentration term. The obtained formulas reproduce free Ca(2+) levels up to 50 microM and their changes in the millisecond range. Derived equations can be useful to predict spatiotemporal profiles of large-amplitude [Ca(2+)] transients, which participate in various physiological processes.

Designing an enzymatic oscillator: Bistability and feedback controlled oscillations with glucose oxidase in a continuous flow stirred tank reactor

The reaction of glucose with ferricyanide catalyzed by glucose oxidase from Aspergillus niger gives rise to a wide range of bistability as the flow rate is varied in a continuous flow stirred tank reactor. Oscillations in pH can be obtained by introducing a negative feedback on the autocatalytic production of H+ that drives the bistability. In our experiments, this feedback consists of an inflow of hydroxide ion at a rate that depends on [H+] in the reactor as k0[OH-]0[H+]/(K+[H+]). pH oscillations are found over a broad range of enzyme and ferricyanide concentrations, residence times (k0 (-1)), and feedback parameters. A simple mathematical model quantitatively accounts for the experimentally found oscillations.

Primary root growth: a biophysical model of auxin-related control

Plant hormones control many aspects of plant development and play an important role in root growth. Many plant reactions, such as gravitropism and hydrotropism, rely on growth as a driving motor and hormones as signals. Thus, modelling the effects of hormones on expanding root tips is an essential step in understanding plant roots. Here we achieve a connection between root growth and hormone distribution by extending a model of root tip growth, which describes the tip as a string of dividing and expanding cells. In contrast to a former model, a biophysical growth equation relates the cell wall extensibility, the osmotic potential and the yield threshold to the relative growth rate. This equation is used in combination with a refined hormone model including active auxin transport. The model assumes that the wall extensibility is determined by the concentration of a wall enzyme, whose production and degradation are assumed to be controlled by auxin and cytokinin. Investigation of the effects of auxin on the relative growth rate distribution thus becomes possible. Solving the equations numerically allows us to test the reaction of the model to changes in auxin production. Results are validated with measurements found in literature.