A numerical study of oxygen diffusion in a spherical cell with the Michaelis-Menten oxygen uptake kinetics

A numerical method is presented for the solution of reaction diffusion systems in biology. The method is used to re-examine the oxygen diffusion in a spherical cell with the Michaelis-Menten oxygen uptake kinetics.     

Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics

The oxygen diffusion in a spherical cell is analyzed in the present work. An oxygen uptake kinetics of the Michaelis-Menten type is employed. The oxygen uptake kinetics predicts the oxygen uptake rates which agree fairly well with the observed data. It has been found that difference between the predicted steady state oxygen tension distribution using the previous simplified oxygen uptake kinetics and that using the present non-linear kinetics is very significant.     

A re-examination of oxygen diffusion in a spherical cell with michaelis-menten oxygen uptake kinetics

Oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is re-examined and the results of a recent paper by Lin corrected. An extension of this model to include external diffusion is made and its effect is shown to be significant. A model which attempts to model the nucleus as a central sphere which does not consume any oxygen is also investigated. Finally, a perturbation solution for a small Michaelis constant is developed.     

Complementary variational principles for diffusion problems with Michaelis-Menten kinetics

The theory of complementary variational principles is used to obtain maximum and minimum principles for diffusion problems with Michaelis-Menten kinetics. In an illustrative calculation we obtain an extremely accurate variational solution in good agreement with the numerical solution of McElwain (1978).     

The kinetics of diffusion between a spherical cell and a surrounding medium with different diffusion properties

The diffusion equation according to Fick’s law is solved for a spherical cell, surrounded by an infinite medium with different diffusion properties. The method of Laplace transform is used to obtain the formal solution, however, no inversion can be found for all times and an expansion suitable for small times is performed. The final expression found is expanded further to be more suited for the determination of the diffusion coefficient from an experimental curve. Application to a biological problem is dissussed.     

A diffusion Michaelis-Menten mechanism: continuous conformational change in enzymatic kinetics

We present a simple model which extends the Michaelis-Menten mechanism by incorporating a continuous protein conformational change in enzymatic catalysis. This model can represent a quantitative version for "rack" or "induced fit" mechanisms. In the steady-state it leads to an equation of the Michaelis-Menten form, but with the catalytic step at the active site showing strong dependence on solvent viscosity. We suggest that a careful examination of solvent viscosity effects on enzymatic activity may serve as a test for the conformational change hypothesis.     

Analysis of a nonlinear diffusion problem With Michaelis-Menten kinetics by an integral equation method

A mathematical model for oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is analyzed by means of an intergral equation method. It is shown that an integral equation formulation can be used to obtain a numerical solution associated with this boundary and initial value problem. Through an illustrative numerical calculation we are able to obtain an accurate solution for both the steady and transient problems. Finally, a comparison is made with the numerical solution of McElwain and the variational solution of Anderson and Arthurs for the steady state and Lin's result concerning the unsteady state.     

Upper and lower bounds from the maximum principle. Intracellular diffusion with Michaelis-Menten kinetics

Analytical bounding functions for diffusion problems with Michaelis-Menten kinetics were recently presented by Anderson and Arthurs, 1985 (Bull. math. Biol. 47, 145–153). Their methods, successful to some extent for a small range of parameters, has the disadvantage of providing a weak upper bound. The optimal approach for the use of one-line bounding kinetics is presented. The use of two-line bounding kinetics is also shown, in order to give, sufficient accuracy in those cases where the one-line approach does not provide satisfactory results. The bounding functions provide excellent upper and lower bounds on the true solution for the entire range of kinetic and transport parameters.     

Morning-to-evening differences in oxygen uptake kinetics in short-duration cycling exercise

This study analyzed diurnal variations in oxygen (O(2)) uptake kinetics and efficiency during a moderate cycle ergometer exercise. Fourteen physically active diurnally active male subjects (age 23+/-5 yrs) not specifically trained at cycling first completed a test to determine their ventilatory threshold (T(vent)) and maximal oxygen consumption (VO(2max)); one week later, they completed four bouts of testing in the morning and evening in a random order, each separated by at least 24 h. For each period of the day (07:00-08:30 h and 19:00-20:30 h), subjects performed two bouts. Each bout was composed of a 5 min cycling exercise at 45 W, followed after 5 min rest by a 10 min cycling exercise at 80% of the power output associated with T(vent). Gas exchanges were analyzed breath-by-breath and fitted using a mono-exponential function. During moderate exercise, the time constant and amplitude of VO(2) kinetics were significantly higher in the morning compared to the evening. The net efficiency increased from the morning to evening (17.3+/-4 vs. 20.5+/-2%; p<0.05), and the variability of cycling cadence was greater during the morning than evening (+34%; p<0.05). These findings suggest that VO(2) responses are affected by the time of day and could be related to variability in muscle activity pattern.     

Buffered diffusion around a spherical proton pumping cell: a theoretical analysis

H⁺ ions are a substrate of many active and passive membrane transporters in all cells. Absolute proton fluxes are often quantified using intracellular pH sensitive microelectrodes or pH sensitive dyes. These measurements, however, rely on a priori estimates of the intracellular buffer capacity and on the assumption of diffusive equilibrium inside the cell. Here, assuming local equilibrium of protons with a single mobile buffer, we model the diffusion of H⁺ in the extracellular medium around an H⁺ pumping cell to estimate the expected pH changes as a function of time, distance from the cell, extracellular buffer capacity, and the absolute proton flux across the membrane. In particular, using accurate numerical simulation, we gauge the range of validity of an explicit, analytical solution of the linearized, nonstationary diffusion equation. Our results provide a framework to quantify the absolute membrane proton flux, if spatiotemporal information about the extracellular pH change is available, e.g., using imaging of pH dependent fluorescent dyes.     

Growth of Trichosporon cutaneum under oxygen limitation: kinetics of oxygen uptake

The relationship between oxygen concentration and growth rate in the yeast Trichosporon cutaneum was studied. In order to establish the conditions for purely oxygen-limited growth, the cells were first grown in a carbon-limited chemostat, and kinetic parameters determined. The cells were then grown in an oxygen-limited chemostat at different dilution rates yielding different oxygen uptake rates. The steady-state dissolved oxygen tension was found at each dilution rate and the corresponding equilibrium dissolved oxygen tension was found at each dilution rate and the corresponding equilibrium dissolved oxygen concentration determined in the effluent medium. The relationship between oxygen concentration and growth rate followed Monod-type kinetics with an apparent K(O) of 4.38 x 10(-6)M.     

Periodic operation of bioreactors for autocatalytic reactions with Michaelis-Menten kinetics

The performance of an isothermal tubular bioreactor carrying out autocatalytic reactions obeying Michaelis-Menten Kinetics is analyzed for improvement in the average yield of product B. Under steady-state condition, the reactor is shown to exhibit input multiplicities in the yield of B with the mean residence time. Simulation results show that a significant improvement in the average yield of B is obtained under feed substrate concentration cycling. The two values of mean residence time giving identical yield under conventional steady-state operation is shown to give distinctly different behaviour under periodic operation. The lower value of the residence time gives improved average yield of B. The performances of the reactor with power law kinetics and that with the Michaelis-Menten kinetics show distinct average yield under periodic operation even though steady-state operation gives identical yield.     

Examination of oxygen uptake kinetics in decremental load exercise by a numerical computation model

The purpose of this study was to establish a numerical computation model for estimation of oxygen uptake (V(.)O2) kinetics in decremental load exercise (DLE) starting from a work rate (WR) above the ventilatory threshold (>VT). In the model, WR in DLE were separated into several steps (constant load exercise, CLE) of which the durations increased step by step. V(.)O2 kinetics in each step was estimated using an exponential equation, and the sum of VO2 values from all steps at a given time was regarded as simulated V(.)O2 in DLE. In the model, the time constants were set symmetrically in a step VT at onset and offset (tau(off)) of exercise. As a result, simulated V(.)O2 qualitatively, but not quantitatively, approximated measured V(.)O2. Consequently, we incorporated a new model in which a step >VT was subdivided into several parts. Although there was a slight difference quantitatively, the interval of subdivision of 3.0 min and tau(off) of 2.8 min allowed for qualitative approximation. The numerical computation model adopted in this study is useful for estimation of V(.)O2 kinetics during DLE starting from high intensity (>VT).     

Salt stress induces a decrease in the oxygen uptake of soybean nodules and in their permeability to oxygen diffusion

The effects of short-term NaCl-salinity on nodules of soybean ( Glycine max L. cv. Kingsoy) were studied on hydroponically-grown plants. Both acetylene reducing activity (ARA) and nodule respiration (O₂ uptake and CO₂ evolution) were immediately inhibited, and the stimulation of them by rising the external partial pressure of O₂ (pO₂) was diminished by the application of 0.1 M NaCl in the nutrient solution. The permeability of the nodule to O₂ diffusion, estimated by O₂ consumption or CO₂ evolution, was significantly lower in the stressed nodules than in the cootrol ones. The respiratory quotient of intact nodules and the ethanol production of excised nodules were increased by low pO₂ and by salt stress. These data confirm that in salt-stressed soybean nodules, O₂ availability is reduced and fermentative pathways are stimulated.     

Analysis of lateral diffusion from a spherical cell surface to a tubular projection.

Cell surfaces are often heterogeneous with respect to the lateral distribution and mobility of membrane components. Because lateral mobility is related to membrane structure, measurement of a particular component's local diffusion coefficient within a distinct surface region provides useful information about the formation and maintenance of that region. Many structurally interesting cell surface features can be described as narrow tubular projections from the body of the cell. In a companion paper, we consider the thin "tethers" that can be mechanically drawn from the red blood cell membrane, and we measure the transport of fluorescent integral proteins from the surface of the cell body onto the tether. In this paper we present an analysis to describe the surface diffusion of membrane particles from a spherical shell onto a thin cylindrical process. Provision is made for different rates of diffusion within the two morphologically distinct regions. The relative role of each region in controlling the diffusive flux between regions is determined primarily by a single dimensionless parameter. This parameter incorporates the ratio of the two diffusion coefficients as well as the dimensions of each region. The analysis can be applied to a fluorescence photobleaching experiment in which the extended process is bleached. If the dimensions of the spherical cell body and the cylindrical extension are known, then the diffusion coefficients of both regions can be determined from the experimental fluorescence recovery curve.     

Mathematical modelling of dynamics and control in metabolic networks. I. On Michaelis-Menten kinetics

As a starting point for modeling of metabolic networks this paper considers the simple Michaelis-Menten reaction mechanism. After the elimination of diffusional effects a mathematically intractable mass action kinetic model is obtained. The properties of this model are explored via scaling and linearization. The scaling is carried out such that kinetic properties, concentration parameters and external influences are clearly separated. We then try to obtain reasonable estimates for values of the dimensionless groups and examine the dynamic properties of the model over this part of the parameter space. Linear analysis is found to give excellent insight into reaction dynamics and it also gives a forum for understanding and justifying the two commonly used quasi-stationary and quasi-equilibrium analyses. The first finding is that there are two separate time scales inherent in the model existing over most of the parameter space, and in particular over the regions of importance here. Full modal analysis gives a new interpretation of quasi-stationary analysis, and its extension via singular perturbation theory, and a rationalization of the quasi-equilibrium approximation. The new interpretation of the quasi-steady state assumption is that the applicability is intimately related to dynamic interactions between the concentration variables rather than the traditional notion that a quasi-stationary state is reached, after a short transient period, where the rates of formation and decomposition of the enzyme intermediate are approximately equal. The modal analysis reveals that the generally used criterion for the applicability of quasi-stationary analysis that total enzyme concentration must be much less than total substrate concentration, et much less than St, is incomplete and that the criterion et much less than Km much less than St (Km is the well known Michaelis constant) is the appropriate one. The first inequality (et much less than Km) guarantees agreement over the longer time scale leading to quasi-stationary behavior or the applicability of the zeroth order outer singular perturbation solution but the second half of the criterion (Km much less than St) justifies zeroth order inner singular perturbation solution where the substrate concentration is assumed to be invariant. Furthermore linear analysis shows that when a fast mode representing the binding of substrate to the enzyme is fast it can be relaxed leading to the quasi-equilibrium assumption. The influence of the dimensionless groups is ascertained by integrating the equations numerically, and the predictions made by the linear analysis are found to be accurate.(ABSTRACT TRUNCATED AT 400 WORDS)