Mariya Ptashnyk

DERIVATION OF A MACROSCOPIC RECEPTOR-BASED MODEL USING HOMOGENIZATION TECHNIQUES

We study the problem of diffusive transport of biomolecules in the intercellular space, modeled as porous medium, and of their binding to the receptors located on the surface membranes of the cells. Cells are distributed periodically in a bounded domain. To describe this process we introduce a reaction-diffusion equation coupled with nonlinear ordinary differential equations on the boundary. We prove existence and uniqueness of the solution of this problem. We consider the limit, when the number of cells tends to infinity and at the same time their size tends to zero, while the volume fraction of the cells remains fixed. Using the homogenization technique of two-scale convergence, we show that the sequence of solutions of the original problem converges to the solution of the so-called macroscopic problem. To show the convergence of the nonlinear terms on the surfaces we use the unfolding method (periodic modulation). We discuss applicability of the result to mathematical description of membrane receptors ...

A dynamic model of nutrient uptake by root hairs.

Root hairs are known to be important in the uptake of sparingly soluble nutrients by plants, but quantitative understanding of their role in this is weak. This limits, for example, the breeding of more nutrient-efficient crop genotypes. We developed a mathematical model of nutrient transport and uptake in the root hair zone of single roots growing in soil or solution culture. Accounting for root hair geometry explicitly, we derived effective equations for the cumulative effect of root hair surfaces on uptake using the method of homogenization. Analysis of the model shows that, depending on the morphological and physiological properties of the root hairs, one of three different effective models applies. They describe situations where: (1) a concentration gradient dynamically develops within the root hair zone; (2) the effect of root hair uptake is negligibly small; or (3) phosphate in the root hair zone is taken up instantaneously. Furthermore, we show that the influence of root hairs on rates of phosphate uptake is one order of magnitude greater in soil than solution culture. The model provides a basis for quantifying the importance of root hair morphological and physiological properties in overall uptake, in order to design and interpret experiments in different circumstances.

BOUNDEDNESS OF SOLUTIONS OF A HAPTOTAXIS MODEL

In this paper we prove the existence of global solutions of the haptotaxis model of cancer invasion for arbitrary non-negative initial conditions. Uniform boundedness of the solutions is shown using the method of bounded invariant rectangles applied to the reformulated system of reaction-diffusion equations in divergence form with a diagonal diffusion matrix. Moreover, the analysis of the model shows how the structure of kinetics of the model is related to the growth properties of the solutions and how this growth depends on the ratio of the sensitivity function (describing the size of haptotaxis) and the diffusion coefficient. One of the implications of our analysis is that in the haptotaxis model with a logistic growth term, cell density may exceed the carrying capacity, which is impossible in the classical logistic equation and its reaction-diffusion extension.

Enhanced zinc uptake by rice through phytosiderophore secretion: a modelling study.

Rice (Oryza sativa L.) secretes far smaller amounts of metal-complexing phytosiderophores (PS) than other grasses. But there is increasing evidence that it relies on PS secretion for its zinc (Zn) uptake. After nitrogen, Zn deficiency is the most common nutrient disorder in rice, affecting up to 50% of lowland rice soils globally. We developed a mathematical model of PS secretion from roots and resulting solubilization and uptake of Zn, allowing for root growth, diurnal variation in secretion, decomposition of the PS in the soil, and the transport and interaction of the PS and Zn in the soil. A sensitivity analysis showed that with realistic parameter values for rice in submerged soil, the typically observed rates of PS secretion from rice are sufficient and necessary to explain observed rates of Zn uptake. There is little effect of diurnal variation in secretion on cumulative Zn uptake, irrespective of other model parameter values, indicating that the observed diurnal variation is not causally related to Zn uptake efficiency. Rooting density has a large effect on uptake per unit PS secretion as a result of overlap of the zones of influence of neighbouring roots. The effects of other complications in the rice rhizosphere are discussed.

Modelling in vitro growth of dense root networks

Hairy roots are plants genetically transformed by Agrobacterium rhizogenes, which do not produce shoots and are composed mainly by roots. Hairy roots of Ophiorrhiza mungos Linn. are currently gaining interest of pharmacologists, since a secondary product of their metabolism, camptothecin, is used in chemotherapy. To optimize the production of valuable secondary metabolites it is necessary to understand the metabolism and growth of these roots systems. In this work, a mathematical model for description of apical growth of a dense root network (e.g. hairy roots) is derived. A continuous approach is used to define densities of root tips and root volume. Equations are posed to describe the evolution of these and are coupled to the distribution of nutrient concentration in the medium and inside the network. Following the principles of irreversible thermodynamics, growth velocity is defined as the sum over three different driving forces: nutrient concentration gradients, space gradients and root tip diffusion. A finite volume scheme was used for the simulation and parameters were chosen to fit experimental data from O. mungos Linn. hairy roots. Internal nutrient concentration determines short-term growth. Long-term behavior is limited by the total nutrient amount in the medium. Therefore, mass yield could be increased by guaranteeing a constant supply of nutrients. Increasing the initial mass of inoculation did not result in higher mass yields, since nutrient consumption due to metabolism also rose. Four different growth strategies are compared and their properties discussed. This allowed to understand which strategy might be the best to increase mass production optimally. The model is able to describe very well the temporal evolution of mass increase and nutrient uptake. Our results provide further understanding of growth and density distribution of hairy root network and therefore it is a sound base for future applications to describe, e.g., secondary metabolite production.

Derivation of a Macroscopic Model for Transport of Strongly Sorbed Solutes in the Soil Using Homogenization Theory

In this paper we derive a model for the diffusion of strongly sorbed solutes in soil taking into account diffusion within both the soil fluid phase and the soil particles. The model takes into account the effect of solutes being bound to soil particle surfaces by a reversible nonlinear reaction. Effective macroscale equations for the solute movement in the soil are derived using homogenization theory. In particular, we use the unfolding method to prove the convergence of nonlinear reaction terms in our system. We use the final, homogenized model to estimate the effect of solute dynamics within soil particles on plant phosphate uptake by comparing our double-porosity model to the more commonly used single-porosity model. We find that there are significant qualitative and quantitative differences in the predictions of the models. This highlights the need for careful experimental and theoretical treatment of plant-soil interaction when trying to understand solute losses from the soil.

A multiscale approach to curvature modulated sorting in biological membranes.

Combining different theoretical approaches, curvature modulated sorting in lipid bilayers fixed on non-planar surfaces is investigated. First, we present a continuous model of lateral membrane dynamics, described by a nonlinear PDE of fourth order. We then prove the existence and uniqueness of solutions of the presented model and simulate membrane dynamics using a finite element approach. Adopting a truly multiscale approach, we use dissipative particle dynamics (DPD) to parameterize the continuous model, i.e. to derive a corresponding macroscopic model. Our model predicts that curvature modulated sorting can occur if lipids or proteins differ in at least one of their macroscopic elastic moduli. Gradients in the spontaneous curvature, the bending rigidity or the Gaussian rigidity create characteristic (metastable) curvature dependent patterns. The structure and dynamics of these membrane patterns are investigated qualitatively and quantitatively using simulations. These show that the decomposition time decreases and the stability of patterns increases with enlarging moduli differences or curvature gradients. Presented phase diagrams allow to estimate if and how stable curvature modulated sorting will occur for a given geometry and set of elastic parameters. In addition, we find that the use of upscaled models is imperative studying membrane dynamics. Compared with common linear approximations the system can evolve to different (meta)stable patterns. This emphasizes the importance of parameters and realistic dynamics in mathematical modeling of biological membranes.

Analysis of root growth from a phenotyping data set using a density-based model

Major research efforts are targeting the improved performance of root systems for more efficient use of water and nutrients by crops. However, characterizing root system architecture (RSA) is challenging, because roots are difficult objects to observe and analyse. A model-based analysis of RSA traits from phenotyping image data is presented. The model can successfully back-calculate growth parameters without the need to measure individual roots. The mathematical model uses partial differential equations to describe root system development. Methods based on kernel estimators were used to quantify root density distributions from experimental image data, and different optimization approaches to parameterize the model were tested. The model was tested on root images of a set of 89 Brassica rapa L. individuals of the same genotype grown for 14 d after sowing on blue filter paper. Optimized root growth parameters enabled the final (modelled) length of the main root axes to be matched within 1% of their mean values observed in experiments. Parameterized values for elongation rates were within ±4% of the values measured directly on images. Future work should investigate the time dependency of growth parameters using time-lapse image data. The approach is a potentially powerful quantitative technique for identifying crop genotypes with more efficient root systems, using (even incomplete) data from high-throughput phenotyping systems.

Hopf bifurcation in a gene regulatory network model: Molecular movement causes oscillations

Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper, we analyze a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatio-temporal interactions between the protein and its mRNA in a one-dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behavior and this is the case for our model, shown initially via computational simulations. In order to investigate this behavior more deeply, we undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. Our result has implications for transcription factors such as p53, NF-κB and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer.

Landau--Lifshitz--Slonczewski Equations: Global Weak and Classical Solutions

We consider magnetization dynamics under the influence of a spin-polarized current, given in terms of a spin-velocity field