Mathematical modelling and simulation of aqueous two-phase continuous protein extraction.

A mathematical model has been developed to describe the continuous, steady-state operation of an aqueous two-phase system for protein extraction. The model is based on steady-state mass balances of the main components and phase equilibrium data. Experimental data on the separation of thaumatin from contaminant proteins of an homogenate of E. coli in a PEG4000/Phosphate system was used. The data shows the effect of the presence and absence of NaCl which was used to carry out the extraction of thaumatin into the PEG phase and back into the PO4(-3) phase. Simulation results showing the sensitivity to key process parameters, and the effect of process variables on performance are presented and discussed. The model can be used to predict performance and thus 'robustness' of process conditions as well as predict protein recovery yield and purity. This model can also be used to implement a suitable control strategy to maintain process stability.     

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 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 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.     

Apparent electric charge of protein molecules. Human thyroxine - binding proteins

1. By comparison of electrophoretic mobilities of two different charged particles under the same conditions the net elementary electrostatic charge of one particle could be calculated when the charge of the other is known. 2. The electrophoretic mobility of human thyroxine - binding globulin does not depend upon the concentration of Tris - HCl buffer in the range 0.05 to 0.20 molar. The value of this mobility is 0.078 and 0.083 cm2 vol(-1) hour(-1) at pH 7.0 and 8.6, respectively. 3. The net elementary electrostatic charge of the human thyroxine - binding globulin appears to be approximately 22 negative elementary electrostatic units in mild alkaline solutions.     

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 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 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 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 oxygen transport to tissue

 The equations governing oxygen transport from blood to tissue are presented for a cylindrical tissue compartment, with blood flowing along a co–axial cylindrical capillary inside the tissue. These governing equations take account of: (i) the non–linear reactions between oxygen and haemoglobin in blood and between oxygen and myoglobin in tissue; (ii) diffusion of oxygen in both the axial and radial directions; and (iii) convection of haemoglobin and plasma in the capillary. A non–dimensional analysis is carried out to assess some assumptions made in previous studies. It is predicted that: (i) there is a boundary layer for oxygen partial pressure but not for haemoglobin or myoglobin oxygen saturation close to the inflow boundary in the capillary; (ii) axial diffusion may not be neglected everywhere in the model; (iii) the reaction between oxygen and both haemoglobin and myoglobin may be assumed to be instantaneous in nearly all cases; and (iv) the effect of myoglobin is only significant for tissue with a low oxygen partial pressure. These predictions are validated by solving the full equations numerically and are then interpreted physically. Received: 13 October 2000 / Revised version: 12 June 2001 / Published online: 17 May 2002     

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.