Direction change of plant root growth by the impenetrable boundary

Spatial boundary conditions must be considered when utilizing mathematical modeling of plant root growth in the container or along with the imbedding solid obstacle. Using basic root growth principles and the geometry of the boundary surface, a mathematical model can be designed to keep all root elements inside the container or outside the obstacle without passing through the boundary after the minimum deflection of growth direction, and it is based on the minimum friction between root tips and soil and energy saving principles. Such a mathematical method is used to simulate the spatial distribution of root growth and the morphological architecture of the root system near the boundary. The validity of this model is supported by experimental observations that confirm some typical characteristics predicted by the simulations. This model can be widely used in resolving boundary condition complications where water and nutrients are consumed by plants in a spatially limited or heterogeneous resource field.     

The strain energy function in axial plant growth

A model for axial plant growth is formulated based on conservation of energy. The model derivation assumes that a strain energy function exists to describe the dissipation of potential energy associated with water uptake, mechanical deformation, and biosynthesis during growth. The derivation does not, however, make any further assumption on the mathematical form of this constitutive relation. The model is employed to investigate possible forms of the strain energy function as applied to steady root growth. Solutions of the nonlinear partial differential equations governing growth are given for cases when the third derivative of the strain energy function is >, <, or =0. These three cases encompass a multitude of mathematical forms of the strain energy function. The resulting solutions are compared with the realization of steady axial root growth. The results of this analysis indicate that a quadratic form of the strain energy function best described steady growth. This conclusion is consistent with previous assumptions on the form of constitutive relations for growth, and allows further interpretation on the water relations, mechanical, and biosynthetic energies associated with plant growth.Research support provided by state and federal funds appropriated to the OSU/OARDC. Journal article no. 12–88     

Modeling pollen tube growth: Feeling the pressure to deliver testifiable predictions

The frequency and amplitude of oscillatory pollen tube growth can be altered by changing the osmotic value of the surrounding medium. This has motivated the proposition that the periodic change in growth velocity is caused by changes in turgor pressure. Using mathematical modeling we recently demonstrated that the oscillatory pollen tube growth does not require turgor to change but that this behavior can be explained with a mechanism that relies on changes in the mechanical properties of the cell wall which in turn are caused by temporal variations in the secretion of cell wall precursors. The model also explains why turgor and growth rate are correlated for oscillatory growth with long growth cycles while they seem uncorrelated for oscillatory growth with short growth cycles. The predictions made by the model are testifiable by experimental data and therefore represent an important step towards understanding the dynamics of the growth behavior in walled cells.     

An on-lattice agent-based Monte Carlo model simulating the growth kinetics of multicellular tumor spheroids

PurposeTo develop an on-lattice agent-based model describing the growth of multicellular tumor spheroids using simple Monte Carlo tools.MethodsCells are situated on the vertices of a cubic grid. Different cell states (proliferative, hypoxic or dead) and cell evolution rules, driven by 10 parameters, and the effects of the culture medium are included. About twenty spheroids of MCF-7 human breast cancer were cultivated and the experimental data were used for tuning the model parameters.ResultsSimulated spheroids showed adequate sizes of the necrotic nuclei and of the hypoxic and proliferative cell phases as a function of the growth time, mimicking the overall characteristics of the experimental spheroids. The relation between the radii of the necrotic nucleus and the whole spheroid obtained in the simulations was similar to the experimental one and the number of cells, as a function of the spheroid volume, was well reproduced. The statistical variability of the Monte Carlo model described the whole volume range observed for the experimental spheroids. Assuming that the model parameters vary within Gaussian distributions it was obtained a sample of spheroids that reproduced much better the experimental findings.ConclusionsThe model developed allows describing the growth of in vitro multicellular spheroids and the experimental variability can be well reproduced. Its flexibility permits to vary both the agents involved and the rules that govern the spheroid growth. More general situations, such as, e. g., tumor vascularization, radiotherapy effects on solid tumors, or the validity of the tumor growth mathematical models can be studied.     

Modeling of typical microbial cell growth in batch culture

A mathematical model was developed, based on the time dependent changes of the specific growth rate, for prediction of the typical microbial cell growth in batch cultures. This model could predict both the lag growth phase and the stationary growth phase of batch cultures, and it was tested with the batch growth ofTrichoderma reesei andLactobacillus delbrucckii.     

Mathematical modeling of thrombus growth in mesenteric vessels

Richardson’s phenomenological mathematical model of the thrombi growth in microvessels is extended to describe the realistic features of the phenomenon. The main directions of the generalization of Richardson’s model are: (1) the dependence of platelet activation time on the distance from the injured vessel wall; (2) the non-homogeneity of the platelet distribution in blood flow in the vicinity of the vessel wall; (3) the adequate choice of the phenomenological function describing the dependence of blood velocity on the thrombus size. The generalization of the model corresponds to the main experimental results and theoretical considerations concerning thrombus formation obtained in recent years. The extended model permits to achieve qualitative agreement between model and experimental data.     

A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

Mathematical model and numerical simulation of the cell growth in scaffolds

A scaffold is a three-dimensional matrix that provides a structural base to fill tissue lesion and provides cells with a suitable environment for proliferation and differentiation. Cell-seeded scaffolds can be implanted immediately or be cultured in vitro for a period of time before implantation. To obtain uniform cell growth throughout the entire volume of the scaffolds, an optimal strategy on cell seeding into scaffolds is important. We propose an efficient and accurate numerical scheme for a mathematical model to predict the growth and distribution of cells in scaffolds. The proposed numerical algorithm is a hybrid method which uses both finite difference approximations and analytic closed-form solutions. The effects of each parameter in the mathematical model are numerically investigated. Moreover, we propose an optimization algorithm which finds the best set of model parameters that minimize a discrete l ₂ error between numerical and experimental data. Using the mathematical model and its efficient and accurate numerical simulations, we could interpret experimental results and identify dominating mechanisms.     

A simple model for periodic cyclostat growth of algae

Summary This paper provides a simple model of nutrient limited periodic cyclostat growth for algae.The basic growth function is assumed to be a time dependent variation of the empirical growth equation developed by Droop (1968). The authors also present the relations for n species' cyclostat coexistence and a stability analysis for the model growth equation.The model, although limited in some respects, agrees very well with available experimental data on Euglena gracilis. The significance of the time dependent amplitude functions developed in this study is also discussed.The term used to describe this system is the cyclostat (see Chisholm et al., 1975).     

A cellular automata model of tumor-immune system interactions

We present a hybrid cellular automata-partial differential equation model of moderate complexity to describe the interactions between a growing tumor next to a nutrient source and the immune system of the host organism. The model allows both temporal and two-dimensional spatial evolution of the system under investigation and is comprised of biological cell metabolism rules derived from both the experimental and mathematical modeling literature. We present numerical simulations that display behaviors which are qualitatively similar to those exhibited in tumor-immune system interaction experiments. These include spherical tumor growth, stable and unstable oscillatory tumor growth, satellitosis and tumor infiltration by immune cells. Finally, the relationship between these different growth regimes and key system parameters is discussed.     

A mathematical model for analysis of the cell cycle in cell lines derived from human tumors

The growth of human cancers is characterised by long and variable cell cycle times that are controlled by stochastic events prior to DNA replication and cell division. Treatment with radiotherapy or chemotherapy induces a complex chain of events involving reversible cell cycle arrest and cell death. In this paper we have developed a mathematical model that has the potential to describe the growth of human tumour cells and their responses to therapy. We have used the model to predict the response of cells to mitotic arrest, and have compared the results to experimental data using a human melanoma cell line exposed to the anticancer drug paclitaxel. Cells were analysed for DNA content at multiple time points by flow cytometry. An excellent correspondence was obtained between predicted and experimental data. We discuss possible extensions to the model to describe the behaviour of cell populations in vivo.     

A new extant respirometric assay to estimate intrinsic growth parameters applied to study plasmid metabolic burden

Start‐up phenomena in microbial biokinetic assays are not captured by the most commonly used growth‐related equations. In this study we propose a new respirometric experimental design to estimate intrinsic growth parameters that allow us to avoid these limitations without data omission, separate mathematical treatment, or wake‐up pulses prior to the analysis. Identifiability and sensitivity analysis were performed to confirm the robustness of the new approach for obtaining unique and accurate estimates of growth kinetic parameters. The new experimental design was applied to establish the metabolic burden caused by the carriage of a pWW0 TOL plasmid in the model organism Pseudomonas putida KT2440. The metabolic burden associated was manifested as a reduction in the yield and the specific growth rate of the host, with both plasmid maintenance and the over‐expression of recombinant proteins from the plasmid contributing equally to the overall effect. Biotechnol. Bioeng. 2010;105: 141–149. © 2009 Wiley Periodicals, Inc.     

微生物复合菌剂的制备

【背景】微生物复合菌剂比单一菌剂更能够在土壤中高效、稳定发挥作用促进作物生长,是微生物菌剂发展的趋势,但是目前对构建微生物复合菌剂的研究不够深入。【目的】研制可显著促进水稻生长的微生物复合菌剂,并构建微生物复合菌剂的数学模型。【方法】将解淀粉芽孢杆菌(Bacillus amyloliquefaciens) FH-1与7株植物促生细菌按照生物量1:1的比例复配成微生物复合菌剂,利用水稻盆栽实验筛选高效复合微生物菌剂。利用生化方法对各个菌株的促生特性进行测定。分析植物特征和菌株促生特性间的相关性关系,并据此利用一般线性方程构建复合菌剂的数学模型。【结果】与空白对照CK相比,复合菌剂FN显著提升水稻的苗长20.79%、根长26.67%和鲜重74.84%(P0.05),是7种微生物复合菌剂中综合效果最佳的微生物复合菌剂。偏相关分析结果表明解无机磷能力、产铁载体能力和产ACC脱氨酶能力在促进植物生长过程中可能发挥着更重要的作用。根据菌剂促生特性与植物特征的偏相关性结果,构建了微生物复合菌剂数学模型,预测准确率可达97%以上。【结论】成功研制了高效促进水稻生长的微生物复合菌剂,并构建了微生物复合菌剂的数学模型,可为微生物复合菌剂的研制提供一定的科学指导。     

Occurrence and Treatment of Bone Atrophic Non-Unions Investigated by an Integrative Approach

Recently developed atrophic non-union models are a good representation of the clinical situation in which many non-unions develop. Based on previous experimental studies with these atrophic non-union models, it was hypothesized that in order to obtain successful fracture healing, blood vessels, growth factors, and (proliferative) precursor cells all need to be present in the callus at the same time. This study uses a combined in vivo-in silico approach to investigate these different aspects (vasculature, growth factors, cell proliferation). The mathematical model, initially developed for the study of normal fracture healing, is able to capture essential aspects of the in vivo atrophic non-union model despite a number of deviations that are mainly due to simplifications in the in silico model. The mathematical model is subsequently used to test possible treatment strategies for atrophic non-unions (i.e. cell transplant at post-osteotomy, week 3). Preliminary in vivo experiments corroborate the numerical predictions. Finally, the mathematical model is applied to explain experimental observations and identify potentially crucial steps in the treatments and can thereby be used to optimize experimental and clinical studies in this area. This study demonstrates the potential of the combined in silico-in vivo approach and its clinical implications for the early treatment of patients with problematic fractures.     

Predicting stability of mixed microbial cultures from single species experiments: 1. Phenomenological model

The growth of mixed microbial cultures on mixtures of substrates is a fundamental problem of both theoretical and practical interest. On the one hand, the literature is abundant with experimental studies of mixed-substrate phenomena [T. Egli, The ecological and physiological significance of the growth of heterotrophic microorganisms with mixtures of substrates, Adv. Microbiol. Ecol. 14 (1995) 305-386]. On the other hand, a number of mathematical models of mixed-substrate growth have been analyzed in the last three decades. These models typically assume specific kinetic expressions for substrate uptake and biomass growth rates and their predictions are formulated in terms of parameters of the model. In this work, we formulate and analyze a general mathematical model of mixed microbial growth on mixtures of substitutable substrates. Using this model, we study the effect of mutual inhibition of substrate uptake rates on the stability of the equilibria of the model. Specifically, we address the following question: How much of the dynamics exhibited by two competing species can be inferred from single species data? We provide geometric criteria for stability of various types of equilibria corresponding to non-competitive exclusion, competitive exclusion, and coexistence of two competing species in terms of growth isoclines and consumption curves. A growth isocline is a curve in the plane of substrate concentrations corresponding to the zero net growth of a given species. In [G.T. Reeves, A. Narang, S.S. Pilyugin, Growth of mixed cultures on mixtures of substitutable substrates: The operating diagram for a structured model, J. Theor. Biol. 226 (2004) 143-157], we introduced consumption curves as sets of all possible combinations of substrate concentrations corresponding to balanced growth of a single microbial species. Both types of curves can be obtained in single species experiments.     

A mathematical model for pH patterns in the rhizospheres of growth zones

In the classical model by Nye (1981), the main process for the change in pH across the rhizosphere is assumed to be diffusion. The classical model focuses on the non-growing part of the root and assumes that the distribution of ion fluxes along the root is spatially uniform. We consider the rhizosphere of the growth zone and take into account the root growth rate and spatially varying flux along the root surface. We present both analytical (dimensional analysis) and experimental (computational) evidence of the importance of taking into account the root growth rate. We describe a conceptual and mathematical model to analyse the pH field around the root tip over time. The model is used with published data to show that, for typical growth rates in sandy soil, the pH field becomes steady (independent of time) after 6 h. Dimensional analysis reveals that a version of the Péclet number, related to the quotient of root elongation rate and proton diffusivity, can be used to predict the extent of the rhizosphere and the time required for it to become steady. For Péclet numbers much greater than 1 (soils), the root influences soil pH for distances on the millimetre scale. In contrast, for Péclet numbers much less than one (agar, aqueous solution), the root influences substrate pH for radial distances on the scale of centimetres. We also present some evidence that agar-contact techniques to measure the soil pH may not be appropriate for measuring the millimetre-scale gradients in soil pH.     

Simulation model of soil compaction and root growth

Soil compaction is a widespread cause of reduced plant productivity. If the effects of soil compaction on plant growth are to be reproduced in simulation models, then the processes through which compaction reduces root elongation must be expressed mathematically and then tested against experimental data. The mathematical theory by which these processes may be represented is given in the accompanying article. In this article, the behavior of a simulation model based on this theory is tested against data for root growth and soil gas concentration recorded from soil columns of which the middle layers were compacted to different bulk densities. The model was able to reproduce the failure of the root system to penetrate the compacted middle layer within the period of the experiment when bulk density exceeded 1.55 Mg m^-3. The model also reproduced decreases in O₂ concentrations, and increases in CO₂ concentrations, in the atmospheres of the compacted layer and of the uncompacted layer below it as bulk density of the compacted layer increased. The simulated time course of O₂ and nutrient uptake and of O₂ concentrations in the compacted layer at different depths is presented and its consistency with experimental findings is examined. As part of a larger ecosystem model, this model will be useful in estimating site-specific effects of soil compaction on carbon cycling in agroecosystems.