Concentration wave of a solute in an artery: the influence of curvature

Mass transport and diffusion phenomena in the arterial lumen are studied through a mathematical model. Blood flow is described by the unsteady Navier-Stokes equation and solute dynamics by an advection-diffusion equation, the convective field being provided by the fluid velocity. A linearization procedure over the steady state solution is carried out and an asymptotic analysis is used to study the effect of a small curvature with respect to the straight tube. Analytical and numerical solutions are found: the results show the characteristics of the long wave propagation and the role played by the geometry on the solute distribution and demonstrate the strong influence of curvature induced by the fluid dynamics.     

Numerical bifurcation analysis of delay differential equations arising from physiological modeling

This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of steady state solutions and periodic solutions of systems of delay differential equations with an arbitrary number of fixed, discrete delays. Second, we demonstrate how these methods can be used to obtain insight into complex biological regulatory systems in which interactions occur with time delays: for this, we consider a system of two equations for the plasma glucose and insulin concentrations in a diabetic patient subject to a system of external assistance. The model has two delays: the technological delay of the external system, and the physiological delay of the patient's liver. We compute stability of the steady state solution as a function of two parameters, compare with analytical results and compute several branches of periodic solutions and their stability. These numerical results allow to infer two categories of diabetic patients for which the external system has different efficiency.     

Method for computer simulation of complex continuous cultivation of microorganisms

A numerical method suitable for computer simulation of complex continuous cultivation of microorganisms is described in detail. The method makes possible an iterative solution of a set of nonlinear algebraic equations that represent the steady state mass balances of a chemostat. The continuation algorithm makes it possible to map the dependence of state variables for the whole range of dilution rates. Easy implementation of the method is possible when the computer code written in BASIC language is used. Two examples, first for oxygen limited cultivation, second for anacrobic acetone — ethanol — butanol fermentation, demonstrate the feasibility of the method.     

A stability analysis of the power-law steady state of marine size spectra

This paper investigates the stability of the power-law steady state often observed in marine ecosystems. Three dynamical systems are considered, describing the abundance of organisms as a function of body mass and time: a “jump-growth” equation, a first order approximation which is the widely used McKendrick–von Foerster equation, and a second order approximation which is the McKendrick–von Foerster equation with a diffusion term. All of these yield a power-law steady state. We derive, for the first time, the eigenvalue spectrum for the linearised evolution operator, under certain constraints on the parameters. This provides new knowledge of the stability properties of the power-law steady state. It is shown analytically that the steady state of the McKendrick–von Foerster equation without the diffusion term is always unstable. Furthermore, numerical plots show that eigenvalue spectra of the McKendrick–von Foerster equation with diffusion give a good approximation to those of the jump-growth equation. The steady state is more likely to be stable with a low preferred predator:prey mass ratio, a large diet breadth and a high feeding efficiency. The effects of demographic stochasticity are also investigated and it is concluded that these are likely to be small in real systems.     

Periodic solutions: a robust numerical method for an S-I-R model of epidemics

 We describe and analyze a numerical method for an S-I-R type epidemic model. We prove that it is unconditionally convergent and that solutions it produces share many qualitative and quantitative properties of the solution of the differential problem being approximated. Finally, we establish explicit sufficient conditions for the unique endemic steady state of the system to be unstable and we use our numerical algorithm to approximate the solution in such cases and discover that it can be periodic, just as suggested by the instability of the endemic steady state. Received: 1 September 1995 / Revised version: 30 April 1997     

A Method for Characterizing the Relation between Nutrient Concentration and Flux into Roots of Intact Plants

A method based on the rate of depletion of a nutrient from solution was developed to characterize nutrient flux of plant roots. Nutrient concentration of the solution was measured at a series of time intervals to describe the complete depletion curve. An integrated rate equation, based on a Michaelis-Menten model, was developed and fit to the data of the depletion curve using a least-square procedure. The equation contained values for V_max, the maximum rate of influx; Km, the Michaelis constant; and E, efflux, which were used to describe the relation between solution concentration and net influx rate. Models other than Michaelis-Menten could also be used. The method uses only one plant or group of plants to obtain data over a range of nutrient concentrations, is adapted particularly to the low concentration range, and measures the concentration below which net influx ceases. With this method the plant is in steady state absorption prior to the experiment and continues at this steady state until near the end of the experiment.     

An approximate solution for a transient two-phase stirred tank bioreactor with nonlinear kinetics

The derivation of an approximate solution method for models of a continuous stirred tank bioreactor where the reaction takes place in pellets suspended in a well-mixed fluid is presented. It is assumed that the reaction follows a Michaelis-Menten-type kinetics. Analytic solution of the differential equations is obtained by expanding the reaction rate expression at pellet surface concentration using Taylor series. The concept of a pellet's dead zone is incorporated; improving the predictions and avoiding negative values of the reagent concentration. The results include the concentration expressions obtained for (a) the steady state, (b) the transient case, imposing the quasi-steady-state assumption for the pellet equation, and (c) the complete solution of the approximate transient problem. The convenience of the approximate method is assessed by comparison of the predictions with the ones obtained from the numerical solution of the original problem. The differences are in general quite acceptable.     

Oxygen tension in thein vivo cornea

The steady state distribution of oxygen tension in thein vivo cornea is estimated in the present study by using a nonlinear oxygen consumption rate equation of the Michaelis-Menten type. Such a rate expression is more accurate than the previous simplified versions in predicting the oxygen consumption rate. It is found that for an open eye with or without contact lens, the oxygen tensions predicted previously are in good agreement with these predicted in the present work. However, for a closed eye with or without contact lens, the previous predictions underestimate the oxygen tension.     

A lactate oxygenase from Mycobacterium phlei. II. Spectral characteristics in aerobic and anaerobic reactions

Lactate oxygenase from Mycobacterium phlei (lactate oxidative decarboxylase) catalyzes the oxygenative conversion of l-lactate to acetate, but acts as an l-lactate dehydrogenase under anaerobic conditions, producing a stoichiometric amount of pyruvate.In an effort to obtain further information on the reaction mechanism, a stopped-flow spectrophotometric technique has been applied to both aerobic and anaerobic reactions of the enzyme. The flavin moiety (FMN) of the enzyme proceeds in a highly oxidized state during the steady state at low lactate and high oxygen concentrations. If the aerobic steady state is maintained at high lactate and low oxygen concentrations, a new spectral species, clearly distinguishable from that of the oxidized state, is visible partially. During anaerobic reduction of the flavin moiety with l-lactate, another species with a weak absorption band at a long wavelength region appears. The significance of these findings in terms of the catalytic reaction mechanism is discussed.     

Diffusion exchange between a membrane-bounded sphere and its surrounding

The diffusion equation is solved for a membrane-bounded sphere situated in an infinite medium with different diffusion properties. The formal solution is obtained through Laplace transformation in the time variable. It is not possible to find a closed form solution in terms of analytical functions, and therefore a numerical inversion technique is applied to obtain the final solution. The application on a biological problem is discussed.     

Nutrient diffusion and simple nth-order consumption in regenerative tissue and biocatalytic sensors

This contribution addresses intra-tissue molar density profiles for nutrients, oxygen, growth factors, and other essential ingredients that anchorage-dependent cells require for successful proliferation on biocompatible surfaces. One-dimensional transient and steady state models of the reaction-diffusion equation are solved to correct a few deficiencies in the first illustrative example of diffusion and zeroth-order rates of consumption in tissues with rectangular geometry, as discussed in Ref. [(Griffith and Swartz, 2006) 1]. The functional form of the molar density profile for each species depends on geometry and the magnitude of the species-specific intra-tissue Damk?hler number. The tissue's central core is reactant starved at high consumption rates and low rates of intra-tissue diffusion when the Damk?hler number exceeds its geometry-sensitive critical value. Ideal tissue engineering designs avoid the diffusion-limited regime such that attached cells are exposed to all of the ingredients required for proliferation everywhere within a regenerative matrix. Analytical and numerical molar density profiles that satisfy the unsteady state modified diffusion equation with pseudo-homogeneous n(th)-order rates of intra-tissue consumption (i.e., n=0,1,2) allow one to (i) predict von Kármán-Pohlhausen mass transfer boundary layer thicknesses, measured inward from the external biomaterial surface toward its central core, and, most importantly, (ii) estimate the time required to achieve steady state conditions for regenerative tissue growth and biocatalytic sensing.     

Analysis of propagating pattern in a chemotaxis system

We consider a cell-chemotaxis model which can generate spatially heterogeneous patterns in cell density and chemoattractant. A local perturbation about the uniform steady state propagates across the domain leaving behind a steady state pattern of standing peaks and troughs. We investigate this patterning process analytically and obtain estimates for the pattern wavelength and speed of spread. We compare the analytical results with numerical simulation of the full model systems.     

A Jump-Growth Model for Predator–Prey Dynamics: Derivation and Application to Marine Ecosystems

This paper investigates the dynamics of biomass in a marine ecosystem. A stochastic process is defined in which organisms undergo jumps in body size as they catch and eat smaller organisms. Using a systematic expansion of the master equation, we derive a deterministic equation for the macroscopic dynamics, which we call the deterministic jump-growth equation, and a linear Fokker–Planck equation for the stochastic fluctuations. The McKendrick–von Foerster equation, used in previous studies, is shown to be a first-order approximation, appropriate in equilibrium systems where predators are much larger than their prey. The model has a power-law steady state consistent with the approximate constancy of mass density in logarithmic intervals of body mass often observed in marine ecosystems. The behaviours of the stochastic process, the deterministic jump-growth equation, and the McKendrick–von Foerster equation are compared using numerical methods. The numerical analysis shows two classes of attractors: steady states and travelling waves.     

Numerical bifurcation analysis of the pattern formation in a cell based auxin transport model

Transport models of growth hormones can be used to reproduce the hormone accumulations that occur in plant organs. Mostly, these accumulation patterns are calculated using time step methods, even though only the resulting steady state patterns of the model are of interest. We examine the steady state solutions of the hormone transport model of Smith et al. (Proc Natl Acad Sci USA 103(5):1301–1306, 2006) for a one-dimensional row of plant cells. We search for the steady state solutions as a function of three of the model parameters by using numerical continuation methods and bifurcation analysis. These methods are more adequate for solving steady state problems than time step methods. We discuss a trivial solution where the concentrations of hormones are equal in all cells and examine its stability region. We identify two generic bifurcation scenarios through which the trivial solution loses its stability. The trivial solution becomes either a steady state pattern with regular spaced peaks or a pattern where the concentration is periodic in time.     

Classification and stability of global inhomogeneous solutions of a macroscopic model of cell motion

Many micro-organisms use chemotaxis for aggregation, resulting in stable patterns. In this paper, the amoeba Dictyostelium discoideum serves as a model organism for understanding the conditions for aggregation and classification of resulting patterns. To accomplish this, a 1D nonlinear diffusion equation with chemotaxis that models amoeba behavior is analyzed. A classification of the steady state solutions is presented, and a Lyapunov functional is used to determine conditions for stability of inhomogenous solutions. Changing the chemical sensitivity, production rate of the chemical attractant, or domain length can cause the system to transition from having an asymptotic steady state, to having asymptotically stable single-step solution and multi-stepped stable plateau solutions.     

Microbial electrode BOD sensors

Two different types of biochemical oxygen demand (BOD) sensors using microbial electrodes were prepared. First, a microbial electrode using the bacteria–collagen membrane and oxygen electrode was used for the determination of BOD. When the electrode was inserted in a sample solution containing glucose and glutamic acid (model waste water), the current of the electrode decreased markedly with time until a steady state was reached. A linear relationship was observed between the steady state current and the concentration of the standard solution containing glucose–glutamic acid or the BOD of the solution. The BOD of industrial waste waters can be estimated within 15 min by using the microbial electrode. No decrease in current output was observed over a ten day period. The reproducibility was determined using the same sample (10% of the standard solution) and was found to be 26.2 ± 2.0 μA (7.5% of the relative standard deviation). Next, a biofuel cell utilizing microbial electrode (immobilized Clostridium butyricum–platinum electrode) was applied to the estimation of the BOD of waste waters. The current of the biofuel cell was decreased markedly with time until a steady state was reached. The steady state current was in all cases attained within 30–40 min at 37°C. A linear relationship was obtained between the steady state current and BOD. The BOD of industrial waste waters can be estimated by using the biofuel cell. Relative error of the BOD estimation was within ±10%. The current output of the biofuel cell was almost constant for 30 days.     

Oxygen diffusion in large single-celled organisms

The functional relationship between the oxygen uptake rate of a spherical, single cell organism and the external oxygen tension is shown to be related to the dependence of the specific oxygen consumption rate, that is, the consumption rate of an infinitesimal volume element of cellular material, on the external oxygen tension. Analytical solutions of the governing steady state diffusion equation are obtained by dividing the system into three regions, an inner region of the sphere in which oxygen consumption rate depends upon oxygen tension, an outer region of the sphere in which oxygen consumption rate is constant (independent of oxygen tension), a nonconsuming membrane over the sphere that offers only resistance to oxygen diffusion, and an infinite region outside the sphere and membrane supplying oxygen to the system. The solutions show the oxygen tension as a function of position inside the spherical cell for a variety of system parameters.     

Viral dynamics model with CTL immune response incorporating antiretroviral therapy

We present two HIV models that include the CTL immune response, antiretroviral therapy and a full logistic growth term for uninfected $\text{ CD4}^+$ T-cells. The difference between the two models lies in the inclusion or omission of a loss term in the free virus equation. We obtain critical conditions for the existence of one, two or three steady states, and analyze the stability of these steady states. Through numerical simulation we find substantial differences in the reproduction numbers and the behaviour at the infected steady state between the two models, for certain parameter sets. We explore the effect of varying the combination drug efficacy on model behaviour, and the possibility of reconstituting the CTL immune response through antiretroviral therapy. Furthermore, we employ Latin hypercube sampling to investigate the existence of multiple infected equilibria.     

On the dynamics of a simple biochemical control circuit

The quantitative dynamics of a biochemical control circuit that regulates enzyme or protein synthesis by end-product feedback is analyzed. We first study a simplified repressible system, which is known to exhibit either a steady state or an oscillatory solution. By showing the analogy of thisn-dimensional system with a time-delay equation for a single variable the mechanism of the self-sustained oscillations becomes transparent. In a more sophisticated system we will find as well either steady state or oscillatory solutions. We determine the role of the parameters with respect to stability and frequency. The most general case will be treated by means of the concept of Lyapunov exponents.