Mathematical modelling of the Warburg effect in tumour cords

The model proposed here links together two approaches to describe tumours: a continuous medium to describe the movement and the mechanical properties of the tissue, and a population dynamics approach to represent internal genetic inhomogeneity and instability of the tumour. In this way one can build models which cover several stages of tumour progression. In this paper we focus on describing transition from aerobic to purely glycolytic metabolism (the Warburg effect) in tumour cords. From the mathematical point of view this model leads to a free boundary problem where domains in contact are characterized by different sets of equations. Accurate stitching of the solution was possible with a modified ghost fluid method. Growth and death of the cells and uptake of the nutrients are related through ATP production and energy costs of the cellular processes. In the framework of the bi-population model this allowed to keep the number of model parameters relatively small.     

数学模型与自然保护科学

日益加剧的人类干扰和景观破碎化已危及全球的生物多样性。自然保护成为人类所面临的最重要也最富有挑战性的任务。指导这一实践的理论和原则极为需要。本文试图综述与自然保护科学有关的几个学科在理论和实际研究(尤其是模型)方面的近期成果以及发展趋势,从而提出自然保护模型的发展方向。文中涉猎基于不同方法论、不同组织水平的模型,并对数学模型在自然保护科学中的作用和实用性加以讨论。     

Mathematical modelling of arsenic transport,distribution and detoxification processes in yeast

Arsenic has a dual role as causative and curative agent of human disease. Therefore, there is considerable interest in elucidating arsenic toxicity and detoxification mechanisms. By an ensemble modelling approach, we identified a best parsimonious mathematical model which recapitulates and predicts intracellular arsenic dynamics for different conditions and mutants, thereby providing novel insights into arsenic toxicity and detoxification mechanisms in yeast, which could partly be confirmed experimentally by dedicated experiments. Specifically, our analyses suggest that: (i) arsenic is mainly protein‐bound during short‐term (acute) exposure, whereas glutathione‐conjugated arsenic dominates during long‐term (chronic) exposure, (ii) arsenic is not stably retained, but can leave the vacuole via an export mechanism, and (iii) Fps1 is controlled by Hog1‐dependent and Hog1‐independent mechanisms during arsenite stress. Our results challenge glutathione depletion as a key mechanism for arsenic toxicity and instead suggest that (iv) increased glutathione biosynthesis protects the proteome against the damaging effects of arsenic and that (v) widespread protein inactivation contributes to the toxicity of this metalloid. Our work in yeast may prove useful to elucidate similar mechanisms in higher eukaryotes and have implications for the use of arsenic in medical therapy.     

Mathematical modelling of aliphatic glucosinolate chain length distribution in Arabidopsis thaliana leaves

Aliphatic glucosinolates are a major class of defensive secondary metabolites in plants that are mostly derived from methionine. Occurring in different chain lengths, they show a structural diversity arising from the variable number of chain elongation cycles taking place during their biosynthesis. The key enzymes in determining glucosinolate chain length are the methylthioalkylmalate (MAM) synthases, MAM1 and MAM3, with MAM3 showing a broader substrate specificity than MAM1. A comparison of the measurements of wild type and MAM1 knockout mutant plants shows the following distinct changes in glucosinolate chain length profiles:

Mathematical modelling of metabolism

Mathematical models of the cellular metabolism have a special interest within biotechnology. Many different kinds of commercially important products are derived from the cell factory, and metabolic engineering can be applied to improve existing production processes, as well as to make new processes available. Both stoichiometric and kinetic models have been used to investigate the metabolism, which has resulted in defining the optimal fermentation conditions, as well as in directing the genetic changes to be introduced in order to obtain a good producer strain or cell line. With the increasing availability of genomic information and powerful analytical techniques, mathematical models also serve as a tool for understanding the cellular metabolism and physiology.     

Mathematical modelling of stress in the hip during gait.

A mathematical model is developed for calculating the contact stress distribution in the hip for a known resultant hip force and characteristic geometrical parameters. Using a relatively simple single nonlinear algebraic equation, the model can be readily applied in clinical practice to estimate the stress distribution in the most frequent body positions of everyday activities. This is demonstrated by analyzing the data on the resultant hip force obtained from laboratory observations where a stance period of gait is considered.     

Mathematical modelling of tumour acidity

Acid-mediated tumour invasion is receiving increasing experimental and clinical attention. Previous models proposed to describe this phenomenon failed to capture key properties of the system, such as the existence of the benign steady state, or predicted incorrectly the size of the inter-tissue gap. Here we show that taking proper account of quiescence ameliorates these drawbacks as well as revealing novel behaviour. The simplicity of the model allows us to fully identify the key parameters controlling different aspects of behaviour.     

Mathematical modelling of the mitochondrial apoptosis pathway

Mitochondria are pivotal for cellular bioenergetics, but are also a core component of the cell death machinery. Hypothesis-driven research approaches have greatly advanced our understanding of the role of mitochondria in cell death and cell survival, but traditionally focus on a single gene or specific signalling pathway at a time. Predictions originating from these approaches become limited when signalling pathways show increased complexity and invariably include redundancies, feedback loops, anisotropies or compartmentalisation. By introducing methods from theoretical chemistry, control theory, and biophysics, computational models have provided new quantitative insights into cell decision processes and have led to an increased understanding of the key regulatory principles of apoptosis. In this review, we describe the currently applied modelling approaches, discuss the suitability of different modelling techniques, and evaluate their contribution to the understanding of the mitochondrial apoptosis pathway. This article is part of a Special Issue entitled Mitochondria: the deadly organelle.     

Mathematical modelling of corneal swelling

This paper presents a differential model of the corneal transport system capable of modelling thickness changes in response to osmotic perturbations applied to either limiting membrane. The work is directed towards understanding corneal behaviour in vivo. The model considers the coupled viscous flows within the corneal stroma and across the epithelial and endothelial membranes. The flows within the stroma are established based on transport theory in porous media, while the flows across the membranes are described using the phenomenological equations of irreversible thermodynamics. The ability of the numerical model to reproduce corneal thickness changes in response to endothelial perturbations was tested against available experimental data. The sensitivity of the model to changes in stromal and membrane transport coefficients was examined.     

Mathematical modelling of biofilm structures

The morphology of biofilms received much attention in the last years. Several concepts to explain the development of biofilm structures have been proposed. We believe that biofilm structure formation depends on physical as well as general and specific biological factors. The physical factors (e.g. governing substrate transport) as well as general biological factors such as growth yield and substrate conversion rates are the basic factors governing structure formation. Specific strain dependent factors will modify these, giving a further variation between different biofilm systems. Biofilm formation seems to be primarily dependent on the interaction between mass transport and conversion processes. When a biofilm is strongly diffusion limited it will tend to become a heterogeneous and porous structure. When the conversion is the rate-limiting step, the biofilm will tend to become homogenous and compact. On top of these two processes, detachment processes play a significant role. In systems with a high detachment (or shear) force, detachment will be in the form of erosion, giving smoother biofilms. Systems with a low detachment force tend to give a more porous biofilm and detachment occurs mainly by sloughing. Biofilm structure results from the interplay between these interactions (mass transfer, conversion rates, detachment forces) making it difficult to study systems taking only one of these factors into account. This revised version was published online in August 2006 with corrections to the Cover Date.     

Mathematical modelling of juxtacrine patterning

Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell—cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied by Collier et al. (1996, J. Theor. Biol. 183, 429–446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback mechanism, and numerical simulations confirm this. Our work shows that juxtacrine signalling provides a novel and robust mechanism for the generation of spatial patterns.     

Mathematical modelling of skeletal repair

Tissue engineering offers significant promise as a viable alternative to current clinical strategies for replacement of damaged tissue as a consequence of disease or trauma. Since mathematical modelling is a valuable tool in the analysis of complex systems, appropriate use of mathematical models has tremendous potential for advancing the understanding of the physical processes involved in such tissue reconstruction. In this review, the potential benefits, and limitations, of theoretical modelling in tissue engineering applications are examined with specific emphasis on tissue engineering of bone. A central tissue engineering approach is the in vivo implantation of a biomimetic scaffold seeded with an appropriate population of stem or progenitor cells. This review will therefore consider the theory behind a number of key factors affecting the success of such a strategy including: stem cell or progenitor population expansion and differentiation ex vivo; cell adhesion and migration, and the effective design of scaffolds; and delivery of nutrient to avascular structures. The focus will be on current work in this area, as well as on highlighting limitations and suggesting possible directions for future work to advance health-care for all.     

Mathematical modelling of tissue-engineered angiogenesis

We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay.     

Alcoholic fermentation: modelling based on sole substrate and product measurement

Optimal automatic bioreactor control requires a mathematical model adapted to the potency of reliable sensors. A new relationship describing the kinetic behavior of alcoholic fermentation is discussed. By analogy with chemical kinetics, the biological rate of substrate consumption is related to substrate and product concentration by the following equation: \documentclass{article}\pagestyle{empty}\begin{document}$$r_s = kS;\alpha P;\beta$$\end{document} Using the well known yield relation between product and substrate, it is possible to describe in both batch and continuous cultures the ethanol and sugar concentrations versus time. This pattern has been successfully tested on several fermentations performed by yeasts (S. cerevisiae, S. bayanus, and S. cerevisiae sake) and a bacterium (Z. mobilis). This simple relationship is proposed as a tool for process control alcoholic fermentation.     

Mathematical modelling of the cellular mechanics of plants

The complex mechanical behaviour of plant tissues reflects the complexity of their structure and material properties. Modelling has been widely used in studies of how cell walls, single cells and tissue respond to loading, both externally applied loading and loads on the cell wall resulting from changes in the pressure within fluid-filled cells. This paper reviews what approaches have been taken to modelling and simulation of cell wall, cell and tissue mechanics, and to what extent models have been successful in predicting mechanical behaviour. Advances in understanding of cell wall ultrastructure and the control of cell growth present opportunities for modelling to clarify how growth-related mechanical properties arise from wall polymeric structure and biochemistry.     

Mathematical modelling of the formation of coated vesicles. A theoretical geometrical approach.

The mechanism by which flat hexagonal lattices of clathrin trimers transform into pentagonal/hexagonal spheres remains a mystery. In light of the geometrical nature of this process we have pursued a mathematical approach to the question. Through the geometrical analysis of flat hexagonal lattices we have discovered three possible forms of transformation to introduce curvature into the centre of the lattice: hub-centre transformation; hub-edge transformation; fringe transformation. Hub-edge and fringe transformations are used first to close the lattice while introducing localized curvature at the edges of the lattice. Hub-centre transformation is used after closure to relax the severely localized curvature generated during closure. This scheme not only maximizes the size of the coated vesicle generated, but also minimizes the number of transformations, thus minimizing the energy expended.