The flux-force relations across complex membranes with active transport

Equations are derived for the total material flux, and the total electric current flux, across a complex membrane system with active transport. The equations describe the fluxes as linear functions of forces across the system, and specifically of electrical potential, hydrostatic pressure, chemical potentials, and active transport rates. The equations can be simplified for experimental studies by making one or more of the forces equal to zero. The osmotic pressure difference across a membrane system is shown to be a function of the electrical potential and chemical potential differences and of the active transport rates. The transmembrane potential is shown to be the sum of a diffusion potential and an active transport potential. A simple equation is derived describing the current across a membrane as a linear function of the electrical potential and the active transport rate. Specific examples of the application of the equations to nerve membrane potentials are considered.     

Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.     

Kinetic analysis of enzyme inactivation by an autodecaying reagent

A simple method is described for the determination of both the pseudo-first-order rate constant and the second-order rate constant for enzyme inactivation by a chemical reagent which itself undergoes exponential decay. The validity of this method has been demonstrated in two test cases in which the labile diethyl pyrocarbonate was used to inactivate salicylate hydroxylase and bacterial luciferase.     

Transport Effects on Multiple-Component Reactions in Optical Biosensors

Optical biosensors are often used to measure kinetic rate constants associated with chemical reactions. Such instruments operate in the surface–volume configuration, in which ligand molecules are convected through a fluid-filled volume over a surface to which receptors are confined. Currently, scientists are using optical biosensors to measure the kinetic rate constants associated with DNA translesion synthesis—a process critical to DNA damage repair. Biosensor experiments to study this process involve multiple interacting components on the sensor surface. This multiple-component biosensor experiment is modeled with a set of nonlinear integrodifferential equations (IDEs). It is shown that in physically relevant asymptotic limits these equations reduce to a much simpler set of ordinary differential equations (ODEs). To verify the validity of our ODE approximation, a numerical method for the IDE system is developed and studied. Results from the ODE model agree with simulations of the IDE model, rendering our ODE model useful for parameter estimation.     

Three steady state situation in an open chemical reaction system. I

In an isothermal continuously stirred tank reactor (open chemical reaction system) fed by sulphuric acid solutions of bromate, bromide and cerium)(III) bistability (three steady state situation) is experimentally observed. This remarkable behavior, based on the instability of one steady state, has important consequences for the understanding of excitability and biochemical control mechanisms. The mass-balance equations for the reactor and the chemical mechanism of the reaction are combined into a simple mathematical model. The behavior of the resulting nonlinear differential equations is examined analytically and by a graphical integration procedure (method of isoclines). Using realistic kinetic data, the model shows the same behavior as observed in the experiment.     

The development of a high-order Taylor expansion solution to the chemical rate equation for the simulation of complex biochemical systems

A numerical method for evaluating chemical rate equations ispresented. This method was developed by expressing the systemof coupled, first-degree, ordinary differential chemical rateequations as a single tensor equation. The tensorial rate equationis invariant in form for all reversible and irreversible reactionschemes that can be expressed as first- and second-order reactionsteps, and can accommodate any number of reactive components.The tensor rate equation was manipulated to obtain a simpleformula (in terms of rate constants and initial concentrations)for the power coefficients of the Taylor expansion of the chemicalrate equation. The Taylor expansion formula was used to developa FORTRAN algorithm for analysing the time development of chemicalsystems. A computational experiment was performed with a Michaelis-Mentenscheme in which step size and expansion order (to the 100thterm) were varied; the inclusion of high-order terms of theTaylor expansion was shown to reduce truncation and round-offerrors associated with Runge-Kutta methods and lead to increasedcomputational efficiency. Received on July 14, 1989; accepted on February 28, 1990     

Chemotaxis and chemokinesis in eukaryotic cells: The Keller-Segel equations as an approximation to a detailed model

More than 20 years after its proposal, Keller and Segel's model (1971,J. theor. Biol.,30, 235–248) remains by far the most popular model for chemical control of cell movement. However, before the Keller-Segel equations can be applied to a particular system, appropriate functional forms must be specified for the dependence on chemical concentration of the cell transport coefficients and the chemical degradation rate. In the vast majority of applications, these functional forms have been chosen using simple intuitive criteria. We focus on the particular case of eukaryotic cell movement, and derive an approximation to the detailed model of Sherrattet al. (1993,J. theor. Biol.,162, 23–40). The approximation consists of the Keller-Segel equations, with specific forms predicted for the cell transport coefficients and chemical degradation rate. Moreover, the parameter values in these functional forms can be directly measured experimentally. In the case of the much studied neutrophil-peptide system, we test our approximation using both the Boyden chamber and under-agarose assays. Finally, we show that for other cell-chemical interactions, a simple comparison of time scales provides a rapid check on the validity of our Keller-Segel approximation.     

Kinetics of unfolding and refolding of proteins. I. Mathematical analysis

A mathematical method for the phenomenological analysis of unimolecular reaction kinetics is presented. Starting from the solution to the differential equations of the system, equations have been developed to relate the unknown parameters inherent to any reaction mechanism, such as the microscopic rate constants for individual reaction steps, to the experimental quantities obtained from the rate of change of average physical properties of the system. Provided that kinetic measurements can be made in both forward and reverse directions, all unknown parameters can be evaluated for many simple mechanisms, and for some of them characteristic relations between observable quantities can be established, which can serve as criteria for immediate rejection. The results will be applied in the following papers to experimental studies of denaturation and renaturation of several proteins.     

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.     

Determination of carbon dioxide evolution rate using on-line gas analysis during dynamic biodegradation experiments

Respirometry is a precious tool for determining the activity of microbial populations. The measurement of oxygen uptake rate is commonly used but cannot be applied in anoxic or anaerobic conditions or for insoluble substrate. Carbon dioxide production can be measured accurately by gas balance techniques, especially with an on-line infrared analyzer. Unfortunately, in dynamic systems, and hence in the case of short-term batch experiments, chemical and physical transfer limitations for carbon dioxide can be sufficient to make the observed carbon dioxide evolution rate (OCER) deduced from direct gas analysis very different from the biological carbon dioxide evolution rate (CER).To take these transfer phenomena into account and calculate the real CER, a mathematical model based on mass balance equations is proposed. In this work, the chemical equilibrium involving carbon dioxide and the measured pH evolution of the liquid medium are considered. The mass transfer from the liquid to the gas phase is described, and the response time of the analysis system is evaluated.Global mass transfer coefficients (K(L)a) for carbon dioxide and oxygen are determined and compared to one another, improving the choice of hydrodynamic hypotheses. The equations presented are found to give good predictions of the disturbance of gaseous responses during pH changes.Finally, the mathematical model developed associated with a laboratory-scale reactor, is used successfully to determine the CER in nonstationary conditions, during batch experiments performed with microorganisms coming from an activated sludge system. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 53: 243-252, 1997.     

A chemical kinetics model for microtubule oscillations

A model for microtubule oscillations is presented based on a set of chemical reaction equations. The rate constants for these reactions are largely determined from experimental data. The plots of assembled tubulin and the phase diagram for assembly are compared with the experimental findings and are found to agree quite well. Copyright 1999 Academic Press.     

Calculation of effective diffusivities for biofilms and tissues.

In this study we describe a scheme for numerically calculating the effective diffusivity of cellular systems such as biofilms and tissues. This work extends previous studies in which we developed the macroscale representations of the transport equations for cellular systems based on the subcellular-scale transport and reaction processes. A finite-difference model is used to predict the effective diffusivity of a cellular system on the basis of the subcellular-scale geometry and transport parameters. The effective diffusivity is predicted for a complex three-dimensional structure that is based on laboratory observations of a biofilm, and these numerical predictions are compared with predictions from a simple analytical solution and with experimental data. Our results indicate that, under many practical circumstances, the simple analytical solution can be used to provide reasonable estimates of the effective diffusivity.     

A new interpretation of the Keller-Segel model based on multiphase modelling

In this paper an alternative derivation and interpretation are presented of the classical Keller-Segel model of cell migration due to random motion and chemotaxis. A multiphase modelling approach is used to describe how a population of cells moves through a fluid containing a diffusible chemical to which the cells are attracted. The cells and fluid are viewed as distinct components of a two-phase mixture. The principles of mass and momentum balance are applied to each phase, and appropriate constitutive laws imposed to close the resulting equations. A key assumption here is that the stress in the cell phase is influenced by the concentration of the diffusible chemical. By restricting attention to one-dimensional cartesian geometry we show how the model reduces to a pair of nonlinear coupled partial differential equations for the cell density and the chemical concentration. These equations may be written in the form of the Patlak-Keller-Segel model, naturally including density-dependent nonlinearities in the cell motility coefficients. There is a direct relationship between the random motility and chemotaxis coefficients, both depending in an inter-related manner on the chemical concentration. We suggest that this may explain why many chemicals appear to stimulate both chemotactic and chemokinetic responses in cell populations. After specialising our model to describe slime mold we then show how the functional form of the chemical potential that drives cell locomotion influences the ability of the system to generate spatial patterns. The paper concludes with a summary of the key results and a discussion of avenues for future research.     

Estimation of intrinsic kinetic constants for pore diffusion-limited immobilized enzyme reactions

A simple method is presented that establishes intrinsic rate parameters when slow pore diffusion of substrate limits immobilized enzyme reactions that obey Michaelis-Menten kinetics. The Aris-Bischoff modulus is employed. Data at high substrate concentrations, where the enzyme would be saturated in the absence of diffusion limitation, and at low substrate concentrations, where effectiveness factors are inversely proportional to reaction modulus, are used to determine maximum rate and Michaelis constant, respectively. Because Michaelis-Menten and Langmuir-Hinshelwood kinetics are formally identical, this method may be used to estimate intrinsic rate parameters of many heterogeneous catalysts. The technique is demonstrated using experimental data from the hydrolysis of maize dextrin with diffusion-limited immobilized glucoamylase. This system yields a Michaelis constant of 0.14%, compared to 0.11% for soluble glucoamylase and 0.24% for immobilized glucoamylase free of diffusional effects.     

Synaptic integration of NMDA and non-NMDA receptors in large neuronal network models solved by means of differential equations

Alpha functions are commonly used to describe different receptor channel kinetics (non-NMDA, GABA_A and GABA_B). In this paper we show that they may be represented as solutions to simple ordinary differential equations. This alternative method is compared with the commonly used direct summation of the alpha function conductances in a high-level neuronal circuit model. A parametric study shows that the differential equation method greatly speeds up the previous summation method. The forward Euler method used to solve this differential equation is shown to be accurate for this type of simulation. The modelling of NMDA receptor channel kinetics is also discussed. Received: 18 December 1992/Accepted in revised form: 28 June 1993     

Synaptic Vesicle Dynamics: A Simple Model of Phasic Release

We present a simple model of phasic neurotransmitter release whichreproduces the salient features of chemical neurotransmission. The synapticvesicle cycle has been modelled as a set of biochemical reactionsrepresented by a system of coupled differential equations. These equationshave been solved analytically to obtain the time dependent behaviour of thesystem on perturbation from the steady state. The scheme of the synapticvesicle network has been emphasized and its role in determining some of themajor experimentally observed properties of synaptic transmission has beendiscussed, which includes the biphasic decay of the rate neurotransmitterrelease even under sustained stimulation. Another interesting outcome ofthis theoretical exercise is the saturation of total release with thecalcium dependent rate constant. The theoretically calculated values oftotal release fit very well into a sigmoidal saturating function with afourth order cooperativity exponent similar to the empiricalDodge–Rahamimoff equation. It appears that the synaptic vesiclenetwork itself is responsible for some of the major properties associatedwith chemical neurotransmission.     

Analysis of the distribution of materials within the blood-brain-cerebrospinal fluid system

When material is introduced into the blood-brain-cerebrospinal fluid system, it will distribute throughout the system in a manner dependent upon the geometrical characteristics and the physical and chemical parameters of the system. Since only average values of the concentration of the material are known in each of the parts of the system, and approximation approach is used in analyzing this distribution theoretically. A simplified model is chosen for the system and equations are derived. These equations are then applied to the results of four papers in the literature dealing with the distribution of thiocyanate. A fifth paper dealing with the sulfate extracellular space of brain is also analyzed.