Tiina Roose

Mathematical Models of Avascular Tumor Growth

This review will outline a number of illustrative mathematical models describing the growth of avascular tumors. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modeling of avascular tumor development are outlined together with a list of key questions.

Electrophysiological characterization of membrane disruption by nanoparticles.

Direct contact of nanoparticles with the plasma membrane is essential for biomedical applications such as intracellular drug delivery and imaging, but the effect of nanoparticle association on membrane structure and function is largely unknown. Here we employ a sensitive electrophysiological method to assess the stability of protein-free membranes in the presence of silica nanospheres of different size and surface chemistry. It is shown that all the silica nanospheres permeabilize the lipid bilayers already at femtomolar concentrations, below reported cytotoxic values. Surprisingly, it is observed that a proportion of the nanospheres is able to translocate over the pure-lipid bilayer. Confocal fluorescence imaging of fluorescent nanosphere analogues also enables estimation of the particle density at the membrane surface; a significant increase in bilayer permeability is already apparent when less than 1% of the bilayer area is occupied by silica nanospheres. It can be envisaged that higher concentrations of nanoparticles lead to an increased surface coverage and a concomitant decrease in bilayer stability, which may contribute to the plasma membrane damage, inferred from lactate dehydrogenase release, that is regularly observed in nanotoxicity studies with cell cultures. This biophysical approach gives quantitative insight into nanosphere-bilayer interactions and suggests that nanoparticle-lipid interactions alone can compromise the barrier function of the plasma membrane.

Impact of growth and uptake patterns of arbuscular mycorrhizal fungi on plant phosphorus uptake—a modelling study

In this paper we present a mathematical model for estimating external mycelium growth of arbuscular mycorrhizal fungi and its effect on root uptake of phosphate (P). The model describes P transport in soil and P uptake by both root and fungi on the single root scale. We investigate differences in soil P depletion and overall P influx into a mycorrhizal root by assuming that different spatial regions of mycelia are active in P uptake. When all external hyphae contribute to P uptake, overall uptake is dominated by the fungus and the most effective growth pattern appears to be the one using a high level of anastomosis. The same is true when only the proportion of external hyphae assumed to be active contributes to uptake. When uptake is restricted to the tips, hyphal contribution to overall P uptake is less dominant; the most effective growth pattern appears to be the one characterised by nonlinear branching where branching stops at a given maximal hyphal tip density. Comparison to measured P depletion in the literature suggests that the scenario where active hyphae are contributing to P uptake is likely to fit the data best. These quantitative predictions promote our understanding of the mycorrhizal symbiosis and its role in plant P nutrition.

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.

Mathematical models of plant–soil interaction

In this paper, we set out to illustrate and discuss how mathematical modelling could and should be applied to aid our understanding of plants and, in particular, plant–soil interactions. Our aim is to persuade members of both the biological and mathematical communities of the need to collaborate in developing quantitative mechanistic models. We believe that such models will lead to a more profound understanding of the fundamental science of plants and may help us with managing real-world problems such as food shortages and global warming. We start the paper by reviewing mathematical models that have been developed to describe nutrient and water uptake by a single root. We discuss briefly the mathematical techniques involved in analysing these models and present some of the analytical results of these models. Then, we describe how the information gained from the single-root scale models can be translated to root system and field scales. We discuss the advantages and disadvantages of different mathematical approaches and make a case that mechanistic rather than phenomenological models will in the end be more trustworthy. We also discuss the need for a considerable amount of effort on the fundamental mathematics of upscaling and homogenization methods specialized for branched networks such as roots. Finally, we discuss different future avenues of research and how we believe these should be approached so that in the long term it will be possible to develop a valid, quantitative whole-plant model.

High resolution synchrotron imaging of wheat root hairs growing in soil and image based modelling of phosphate uptake

· Root hairs are known to be highly important for uptake of sparingly soluble nutrients, particularly in nutrient deficient soils. Development of increasingly sophisticated mathematical models has allowed uptake characteristics to be quantified. However, modelling has been constrained by a lack of methods for imaging live root hairs growing in real soils. · We developed a plant growth protocol and used Synchrotron Radiation X-ray Tomographic Microscopy (SRXTM) to uncover the three-dimensional (3D) interactions of root hairs in real soil. We developed a model of phosphate uptake by root hairs based directly on the geometry of hairs and associated soil pores as revealed by imaging. · Previous modelling studies found that root hairs dominate phosphate uptake. By contrast, our study suggests that hairs and roots contribute equally. We show that uptake by hairs is more localized than by roots and strongly dependent on root hair and aggregate orientation. · The ability to image hair-soil interactions enables a step change in modelling approaches, allowing a more realistic treatment of processes at the scale of individual root hairs in soil pores.

Three‐dimensional quantification of soil hydraulic properties using X‐ray Computed Tomography and image‐based modeling

We demonstrate the application of a high-resolution X-ray Computed Tomography (CT) method to quantify water distribution in soil pores under successive reductive drying. We focus on the wet end of the water release characteristic (WRC) (0 to −75 kPa) to investigate changes in soil water distribution in contrasting soil textures (sand and clay) and structures (sieved and field structured) and to determine the impact of soil structure on hydraulic behavior. The 3-D structure of each soil was obtained from the CT images (at a 10 μm resolution). Stokes equations for flow were solved computationally for each measured structure to estimate hydraulic conductivity. The simulated values obtained compared extremely well with the measured saturated hydraulic conductivity values. By considering different sample sizes we were able to identify the smallest possible representative sample size which is required to determine a globally valid hydraulic conductivity.

Verification and intercomparison of reactive transport codes to describe root-uptake

Several mathematical models have been developed to simulate processes and inter- actions in the plant rhizosphere. Most of these models are based on a rather simplified descrip- tion of the soil chemistry and interactions of plant roots in the rhizosphere. In particular the feed- back loops between exudation, water and solute uptake are mostly not considered, although their importance in the bioavailability of mineral ele- ments for plants has been demonstrated. The aim of this work was to evaluate three existing coupled speciation-transport tools to model rhi- zosphere processes. In the field of hydrogeo- chemistry, such computational tools have been developed to describe acid-base and redox reac- tions, complexation and ion exchange, adsorption and precipitation of chemical species in soils and aquifers using thermodynamic and kinetic rela- tionships. We implemented and tested a simple rhizosphere model with three geochemical com- putational tools (ORCHESTRA, MIN3P, and PHREEQC). The first step was an accuracy

Growth model for arbuscular mycorrhizal fungi

In order to quantify the contribution of arbuscular mycorrhizal (AM) fungi to plant phosphorus nutrition, the development and extent of the external fungal mycelium and its nutrient uptake capacity are of particular importance. We develop and analyse a model of the growth of AM fungi associated with plant roots, suitable for describing mechanistically the effects of the fungi on solute uptake by plants. The model describes the development and distribution of the fungal mycelium in soil in terms of the creation and death of hyphae, tip–tip and tip–hypha anastomosis, and the nature of the root–fungus interface. It is calibrated and corroborated using published experimental data for hyphal length densities at different distances away from root surfaces. A good agreement between measured and simulated values was found for three fungal species with different morphologies: Scutellospora calospora (Nicol. & Gerd.) Walker & Sanders; Glomus sp.; and Acaulospora laevis Gerdemann & Trappe associated with Trifolium subterraneum L. The model and findings are expected to contribute to the quantification of the role of AM fungi in plant mineral nutrition and the interpretation of different foraging strategies among fungal species.

Assessing the influence of the rhizosphere on soil hydraulic properties using X-ray computed tomography and numerical modelling

Highlight Using non-destructive imaging techniques and numerical modelling, we quantify differences in hydraulic and structural properties of bulk and rhizosphere soil for sand and clay loam soils.