Mathematical modelling and analysis of cellular signalling in macrophages

Cell signalling pathways play a crucial role in proper cell development and behaviour, with implications to survival, chemotaxis, proliferation, and even programmed cell death known as apoptosis. In this article, we outline a mathematical model of the G-protein signalling pathway in a particular cell line of macrophages, focusing on activation of a particular G-protein-coupled receptor, P2Y(6). The model is based on the kinetics of P2Y(6) surface receptors, inositol trisphosphate, cytosolic calcium, and differential dynamics of multiple species of diacylglycerol. Insight into the dynamics of the system is given through recently available experimental results and incorporated into the model. Mathematical analysis of the model, including establishment of global existence, uniqueness, positivity, and boundedness of solutions, and global stability of a unique steady-state solution is discussed.     

Mathematical modelling in physiology

The article illustrates the method of mathematical modelling in physiology as a unique tool to study physiological processes. A number of demonstrated examples appear as a result of long-term experience in mathematical modelling of electrical and mechanical phenomena in the heart muscle. These examples are presented here to show that the modelling provides insight into mechanisms underlying these phenomena and is capable to predict new ones that were previously unknown. While potentialities of the mathematical modelling are analyzed with regard to the myocardium, they are quite universal to deal with any physiological processes.     

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 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 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.     

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.     

Soy isoflavones and cellular mechanics

Soy isoflavones are diphenolic compounds that are frequently used for alternative treatment of ageing symptoms in both genders. They operate at principally two hierarchical levels of functional organization – cellular and molecular, while these ‘types’ of action appear to have indefinite borders. Soy isoflavone action at the cellular level involves inter alia the effects on cell mechanics. This epigenetic and modular determinant of cell function and fate is defined by: the anchorage to extracellular matrix (ECM) and neighboring cells, cytoskeleton organization, membrane tension and vesicle trafficking. Soy isoflavones have been reported to: (i) generally fashion an inert cell phenotype in some cancers and enhance the cell anchorage in connective tissues, via the effects on ECM proteins, focal adhesion kinases-mediated events and matrix metalloproteinases inhibition; (ii) affect cytoskeleton integrity, the effects being related to Ca²⁺ ions fluxes and involving cell retraction or differentiation/proliferation-related variations in mechanical status; (iii) increase, remain “silent” or decrease membrane tension/fluidity, which depends on polarity and a number and arrangement of functional groups in applied isoflavone; (iv) provoke inhibitory effects on vesicle trafficking and exo-/endocytosis, which are usually followed by changed cell morphology. Here we present and discuss the abundance of effects arising from cells’ “encounter” with soy isoflavones, focusing on different morphofunctional definers of cell mechanics.     

Cardiac mechanics at the cellular level.

The active mechanical properties of heart muscle are load, length, and time-dependent. The capability for investigating cardiac mechanisms at the cellular level may help to distinguish between those properties of the myocardium which arise from myocardial cells and those which arise from the tissue architecture and extracellular matrix of connective fibers. We present here, for the first time, a general approach for subjecting single heart cells to isometric, isotonic, afterloaded, or physiological loading sequences, while obtaining on-line measures of cell force and length. This approach has been implemented and tested on freshly dissociated, adult frog ventricular myocytes. Examples are presented for each of the four loading sequences.     

Mathematical modelling of clostridial acetone-butanol-ethanol fermentation

Applied Microbiology and Biotechnology - Clostridial acetone-butanol-ethanol (ABE) fermentation features a remarkable shift in the cellular metabolic activity from acid formation, acidogenesis, to...     

Mathematical modelling of interwound DNA supercoils

Small loops of DNA are affected by a variety of enzymes which remove turns of twist relative to the underlying double-helical structure. The molecule adopts a complex three-dimensional shape known as a supercoil in order to relieve the resulting internal stresses. This article describes an approach to modelling the overall shape of the supercoiled structure using elastic rod theory, which leads to simple expressions for predicting the shape of the structure. Predictions on the number of crossings in the balanced ply and the length of the end loops are compared to data in the literature and show reasonable agreement. The effect of the charged phosphate groups along the backbone of the DNA on the resulting supercoiled shape are also examined, and it is shown that this shape is very sensitive to the ionic concentration of the solution.     

Mathematical modelling of whole chromosome replication

All chromosomes must be completely replicated prior to cell division, a requirement that demands the activation of a sufficient number of appropriately distributed DNA replication origins. Here we investigate how the activity of multiple origins on each chromosome is coordinated to ensure successful replication. We present a stochastic model for whole chromosome replication where the dynamics are based upon the parameters of individual origins. Using this model we demonstrate that mean replication time at any given chromosome position is determined collectively by the parameters of all origins. Combining parameter estimation with extensive simulations we show that there is a range of model parameters consistent with mean replication data, emphasising the need for caution in interpreting such data. In contrast, the replicated-fraction at time points through S phase contains more information than mean replication time data and allowed us to use our model to uniquely estimate many origin parameters. These estimated parameters enable us to make a number of predictions that showed agreement with independent experimental data, confirming that our model has predictive power. In summary, we demonstrate that a stochastic model can recapitulate experimental observations, including those that might be interpreted as deterministic such as ordered origin activation times.     

Mathematical modelling of pulmonary gas transport

 Equations governing the transport of the gases oxygen and carbon dioxide inside the pulmonary capillaries are written down. By analysing these equations it is predicted that there will be negligible limitation to the transport of oxygen when oxygen concentration takes a normal physiological or higher value. For low values of oxygen concentration, there may be limitation to oxygen transport. It is predicted further that the quantity of carbon dioxide excreted from blood into alveolar gas is dependent on oxygen concentration, with low oxygen concentrations inhibiting the carbon dioxide transport process. The relatively slow reaction involving carbon dioxide in plasma also inhibits the excretion of carbon dioxide. These predictions are verified by solving the whole system of governing equations numerically. Received: 1 May 2002 / Revised version: 20 October 2002 / Published online: 19 March 2003 JPW was supported by a grant from the Engineering and Physical Sciences Research Council of Great Britain. Key words or phrases: Pulmonary gas transport – Haemoglobin – Saturation     

Mathematical modelling of intra-aortic balloon pump

Background: Ischemic heart diseases now afflict thousands of Iranians and are the major cause of death in many industrialised countries. Mathematical modelling of an intra-aortic balloon pump (IABP) could provide a better understanding of its performance and help to represent blood flow and pressure in systemic arteries before and after inserting the pump.

Methods: A mathematical modelling of the whole cardiovascular system was formulated using MATLAB software. The block diagram of the model consists of 43 compartments. All the anatomical data was extracted from the physiological references. In the next stage, myocardial infarction (MI) was induced in the model by decreasing the contractility of the left ventricle. The IABP was mathematically modelled and inserted in the model in the thoracic aorta I artery just before the descending aorta. The effects of IABP on MI were studied using the mathematical model.

Results: The normal operation of the cardiovascular system was studied firstly. The pressure–time graphs of the ventricles, atriums, aorta, pulmonary system, capillaries and arterioles were obtained. The volume–time curve of the left ventricle was also presented. The pressure–time curves of the left ventricle and thoracic aorta I were obtained for normal, MI, and inserted IABP conditions. Model verification was performed by comparing the simulation results with the clinical observations reported in the literature.

Conclusions: IABP can be described by a theoretical model. Our model representing the cardiovascular system is capable of showing the effects of different pathologies such as MI and we have shown that MI effects can be reduced using IABP in accordance with the modelling results. The mathematical model should serve as a useful tool to simulate and better understand cardiovascular operation in normal and pathological conditions.     

Mathematical modelling of the intravenous glucose tolerance test

 Several attempts at building a satisfactory model of the glucose-insulin system are recorded in the literature. The minimal model, which is the model currently mostly used in physiological research on the metabolism of glucose, was proposed in the early eighties for the interpretation of the glucose and insulin plasma concentrations following the intravenous glucose tolerance test. It is composed of two parts: the first consists of two differential equations and describes the glucose plasma concentration time-course treating insulin plasma concentration as a known forcing function; the second consists of a single equation and describes the time course of plasma insulin concentration treating glucose plasma concentration as a known forcing function. The two parts are to be separately estimated on the available data. In order to study glucose-insulin homeostasis as a single dynamical system, a unified model would be desirable. To this end, the simple coupling of the original two parts of the minimal model is not appropriate, since it can be shown that, for commonly observed combinations of parameter values, the coupled model would not admit an equilibrium and the concentration of active insulin in the “distant” compartment would be predicted to increase without bounds. For comparison, a simple delay-differential model is introduced, is demonstrated to be globally asymptotically stable around a unique equilibrium point corresponding to the pre-bolus conditions, and is shown to have positive and bounded solutions for all times. The results of fitting the delay-differential model to experimental data from ten healthy volunteers are also shown. It is concluded that a global unified model is both theoretically desirable and practically usable, and that any such model ought to undergo formal analysis to establish its appropriateness and to exclude conflicts with accepted physiological notions. Received: 22 June 1998 / Revised version: 24 February 1999     

Mathematical modelling of the heat resistance of Listeria monocytogenes

The heat resistance of Listeria monocytogenes phagovar 2389/2425/3274/2671/47/108/340 (1992 French outbreak strain) in broth was studied at 55, 60 and 65 °C. Experiments were carried out on bacterial cultures in three different physiological states: cultures at the end of the log phase, cultures heat-shocked at 42 °C for 1 h, and subcultures of cells resistant to prolonged heating. Survivor curves were better fitted using a sigmoidal equation than the classical log-linear model. This approach was justified by the existence of heat resistance distributions within the bacterial populations. Peaks (log₁₀ of heating time) of heat resistance distributions of untreated, heat-shocked, and selected cultures at 55, 60 and 65 °C were 0·34, −0·90 and −1·84 min, 0·74, −0·51 and −1·24 min, and 0·17, −0·94 and−1·45 min, respectively. The widths of the distributions are proportional to 0·29, 0·36and 0·41 min^0·5, 0·26, 0·36 and 0·41 min^0·5, and 0·34, 0·44 and 0·41 min^0·5. An increase in thethermal tolerance could then be induced by sublethal heat shock or by selection of heatresistant cells.     

Mathematical modelling of the degradation behaviour of biodegradable metals

A mathematical model for the biodegradation of magnesium is developed in this study to inspect the corrosion behaviour of biodegradable implants. The aim of this study was to provide a suitable framework for the assessment of the corrosion rate of magnesium which includes the process of formation/dissolution of the protective film. The model is intended to aid the design of implants with suitable geometries. The level-set method is used to follow the changing geometry of the implants during the corrosion process. A system of partial differential equations is formulated based on the physical and chemical processes that occur at the implant-medium boundary in order to simulate the effect of the formation of a protective film on the degradation rate. The experimental data from the literature on the corrosion of a high-purity magnesium sample immersed in simulated body fluid is used to calibrate the model. The model is then used to predict the degradation behaviour of a porous orthopaedic implant. The model successfully reproduces the precipitation of the corrosion products on the magnesium surface and the effect on the degradation rate. It can be used to simulate the implant degradation and the formation of the corrosion products on the surface of biodegradable magnesium implants with complex geometries.