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.     

Application of catastrophe theory to corneal swelling

Stromal swelling in human, cat, and rabbit cornea is biphasic, interpretable as an elementary cusp catastrophe proposed by Thom, with t* = log t and Q* = log Q (stromal charge Q, time t) as control parameters, and H0.5 (hydration H) as the state variable. A thermodynamic potential with two attractor regions, each with a local minimum, governs corneal stromal swelling. Transitions follow a 'saturation convention' whereby the second minimum is preferred upon availability. Corneal swelling is an example of a space-equivalent unfolding, where the transition plane moves in time. It is proposed that the transition plane coincides with the uncoupling of interfibrillary linkages or 'springs' in the corneal stroma, and is associated with a critical hydration of ca. 10 kg H2O per kilogram dry mass, and stromal charge ca. 1 x 10(-7) mol electrons.     

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 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.     

Numerical modelling of corneal biomechanical behaviour

Numerical modelling based on finite element analysis is used to represent the biomechanical behaviour of the cornea. The construction details of the model including the discretisation method, the mesh density, the thickness distribution, the topography idealisation, the boundary conditions and the material properties, are optimised to improve efficiency. Factors which are found to have a considerable effect on model accuracy are considered and those with effect below a certain low threshold are ignored to reduce cost of analysis. The model is validated against laboratory tests involving pressure inflation of corneal trephinates while monitoring their behaviour. To illustrate the potential of the validated model in studying corneal biomechanics, its use in modelling Goldmann applanation tonometry (GAT) is briefly described. In studying GAT, the model is able to accurately trace the behaviour of the cornea under tonometric pressure and monitor the gap closure and the progress of deformation to the point of applanation.     

Numerical modelling of corneal biomechanical behaviour

Numerical modelling based on finite element analysis is used to represent the biomechanical behaviour of the cornea. The construction details of the model including the discretisation method, the mesh density, the thickness distribution, the topography idealisation, the boundary conditions and the material properties, are optimised to improve efficiency. Factors which are found to have a considerable effect on model accuracy are considered and those with effect below a certain low threshold are ignored to reduce cost of analysis. The model is validated against laboratory tests involving pressure inflation of corneal trephinates while monitoring their behaviour. To illustrate the potential of the validated model in studying corneal biomechanics, its use in modelling Goldmann applanation tonometry (GAT) is briefly described. In studying GAT, the model is able to accurately trace the behaviour of the cornea under tonometric pressure and monitor the gap closure and the progress of deformation to the point of applanation.     

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 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 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 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 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 immunity to malaria

A comparison of two epidemiological models of immunity to malaria shows that different characterizations of immunity boosted by exposure to infection generate qualitatively different results. Attempts to control disease by reducing transmission or increasing the recovery rate can produce an increase in prevalence in the compartmental model with discrete epidemiological states. However, the parasite density always decreases in response to disease control in the model with continuous epidemiological variables. Each model accounts for some epidemiological patterns. The increase in prevalence seen in the compartmental model is in accord with observed effects of variation in transmission. Parasite suppression in areas of antimalarial drug use is consistent with the effect of an increased recovery rate in the density model. Future work should combine the two approaches, perhaps by using the compartmental model over the low to moderate range of infection rates and switching to the density model at high infection rates. In any case, the validation of models needs to take account of the usage of antimalarial drugs as well as the intensity of transmission.     

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。     

A triphasic analysis of corneal swelling and hydration control.

Physiological studies strongly support the view that hydration control in the cornea is dependent on active ion transport at the corneal endothelium. However, the mechanism by which endothelial ion transport regulates corneal thickness has not been elaborated in detail. In this study, the corneal stroma is modeled as a triphasic material under steady-state conditions. An ion flux boundary condition is developed to represent active transport at the endothelium. The equations are solved in cylindrical coordinates for confined compression and in spherical coordinates to represent an intact cornea. The model provides a mechanism by which active ion transport at the endothelium regulates corneal hydration and provides a basis for explaining the origin of the "imbibition pressure" and stromal "swelling pressure." The model encapsulates the Donnan view of corneal swelling as well as the "pump-leak hypothesis."