pH-controlled continuous cultivation of mycoplasmas

The continuous cultivation of mycoplasmas in a pH-controlled metabolistat was investigated with the fermentative strain Mycoplasma mobile 163K and the nonfermentative strain Mycoplasma arthritidis ISR1. The addition of medium and the removal of culture suspension were regulated by acid production from glucose by M. mobile 163K and by ammonium production from arginine by M. arthritidis ISR1, respectively. For both strains the optimal pH for continuous growth was 7.0. The steady state could be maintained for at least 21 days. With CFU of 8.4 X 10(9) ml-1 (M. mobile 163K) and 3.2 X 10(9) ml-1 (M. arthritidis ISR1), the cell concentrations were slightly higher than those obtained in batch cultures. The dependence on the adjusted pH values was measured for several parameters, such as flow rate, CFU, glucose fermentation or production of ammonia, and gliding velocity. Since the long lag phases of batch cultures can be avoided, pH-controlled continuous cultures provide an appropriate system for the production of mycoplasma cells.     

pH-controlled continuous cultivation of mycoplasmas.

The continuous cultivation of mycoplasmas in a pH-controlled metabolistat was investigated with the fermentative strain Mycoplasma mobile 163K and the nonfermentative strain Mycoplasma arthritidis ISR1. The addition of medium and the removal of culture suspension were regulated by acid production from glucose by M. mobile 163K and by ammonium production from arginine by M. arthritidis ISR1, respectively. For both strains the optimal pH for continuous growth was 7.0. The steady state could be maintained for at least 21 days. With CFU of 8.4 X 10(9) ml-1 (M. mobile 163K) and 3.2 X 10(9) ml-1 (M. arthritidis ISR1), the cell concentrations were slightly higher than those obtained in batch cultures. The dependence on the adjusted pH values was measured for several parameters, such as flow rate, CFU, glucose fermentation or production of ammonia, and gliding velocity. Since the long lag phases of batch cultures can be avoided, pH-controlled continuous cultures provide an appropriate system for the production of mycoplasma cells.     

Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications

A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered.     

Harmless delays in model systems

An autonomous system of two differential equations with an asymptotically stable steady state is considered. It is shown that there exists a class of time delays which preserves the stability of such a steady state. The charactristic return time (or decay rate) for the solutions to approach the steady state is estimated using the variational system for the perturbations.     

Theoretical and experimental aspects of erythrocyte filterability testing; flow acceleration and systemic resistance

Blood cell filterability is an established method in blood rheology. The dynamics at flow onset and its relevance to the data interpretation is, however, not fully known. This paper aims to investigate what controls the length and slope of flow acceleration as the medium accelerates to reach the steady state, and how this phenomenon may interfere with the data output. The acceleration time was not constant. With buffer the steady-state flow showed a logarithmic correlation (p<0.05) versus acceleration time and a linear correlation (p<0.001) versus acceleration slope. With 5% erythrocyte resuspension the steady-state flow instead demonstrated a linear relationship versus acceleration time (p<0.001) and no correlation versus acceleration slope. A cut-off timing of 0.6s is suggested to avoid artifacts associated with flow acceleration. The possible influence on data interpretation from the flow channel systemic resistance was also addressed, and found to significantly underestimate measurable changes in erythrocyte properties from unprocessed flow curves. This was despite the traditional correction for blank filtration flow. Both acceleration and effects from systemic resistance do probably have minor influence on the historic data interpretation but could perhaps be considered in the methodology to sharpen the data output.     

The growth of pure and simple microbial competitors in a moving distributed medium.

The dynamics of pure and simple competition between two microbial species are examined for the case of interaction arising in a distributed and nonstagnant environment. The environment is modeled as a tubular reactor. It is shown that for relatively small values of the dispersion coefficient (i.e., for small, but nonzero, backmixing of the medium), the two competing populations can coexist in a stable steady state. It has been assumed that the species grow uninhibited and that if there are maintenance requirements they are satisfied from endogenous sources. From numerical studies it has been found that a necessary condition for coexistence is that the net specific growth rate curves of the two competitors cross each other at a positive value of the concentration of the rate-limiting substrate. The model equations have been numerically solved by using the methods of orthogonal and spline collocation.     

A fully coupled porous media and channels flow approach for simulation of blood and bile flow through the liver lobules

Two dimensional, steady state, and incompressible blood and bile flows through the liver lobules are numerically simulated. Two different geometric models A and B are proposed to study the effects of lobule structure on the fluid flow behaviour. In Model A, the lobule tissue is represented as a hexagonal shape porous medium with a set of flow channels at its vertices accounting for the hepatic artery, portal and central veins along with bile ductules. Model B is a channelized porous medium constructed by adding a set of flow channels, representing the bile canaliculies and lobule sinusoids, to Model A. The bile and blood flow through the lobule is simulated by the finite element approach, based on the Darcy/Brinkman equations in the lobule tissue and the Navier-Stokes (or Stokes) equations in the flow channels. In Model B, a transmission factor on the boundaries of the bile canaliculies is introduced to connect the bile and blood flows. First, a single regular lobule is utilized to exhibit the fluid flow pattern through the liver lobule represented by proposed geometric models. Then, the model is extended to a group of liver lobules to demonstrate the flow through a liver slice represented by irregular lobules. Numerical results indicate that the Darcy and Brinkman equations provide nearly the same solutions for Model A and similar solutions with a little difference for Model B. It is shown that the existence of sinusoids and bile canaliculies inside the liver lobules has noticeable effects on its fluid flow pattern, in terms of pressure and velocity fields.     

Steady state enzyme kinetics for systems with three enzyme-binding species

The steady state enzyme kinetics of those systems are discussed, which involve three species binding to enzymes. Two specific systems are considered. In one system, all three species bind only once to the enzyme. In the other system, two species bind once and one binds twice to the enzyme. The species are labeled S, A and B. The general case is considered, in which all possible complexes involving enzyme E and species S generate product P. Species A and B may become co-substrates, activators or inhibitors. The steady state enzyme kinetic equations for the general case for both systems are presented. These equations are further discussed for a number of special cases, which may be of interest to enzymologists and others using enzymes.     

Modeling the dynamics of Hepatitis C virus with combined antiviral drug therapy: Interferon and Ribavirin

A mathematical modeling of hepatitis C virus (HCV) dynamics and antiviral therapy has been presented in this paper. The proposed model, which involves four coupled ordinary differential equations, describes the interaction of target cells (hepatocytes), infected cells, infectious virions and non-infectious virions. The model takes into consideration the addition of ribavirin to interferon therapy and explains the dynamics regarding a biphasic and triphasic decline of viral load in the model. A critical drug efficacy parameter has been defined and it is shown that for an efficacy above this critical value, HCV is eradicated whereas for efficacy lower this critical value, a new steady state for infectious virions is reached, which is lower than the previous steady state value.     

Population dynamics and competition in chemostat models with adaptive nutrient uptake

 The standard Monod model for microbial population dynamics in the chemostat is modified to take into consideration that cells can adapt to the change of nutrient concentration in the chemostat by switching between fast and slow nutrient uptake and growing modes with asymmetric thresholds for transition from one mode to another. This is a generalization of a modified Monod model which considers adaptation by transition between active growing and quiescent cells. Global analysis of the model equations is obtained using the theory of asymptotically autonomous systems. Transient oscillatory population density and hysteresis growth pattern observed experimentally, which do not occur for the standard Monod model, can be explained by such adaptive mechanism of the cells. Competition between two species that can switch between fast and slow nutrient uptake and growing modes is also considered. It is shown that generically there is no coexistence steady state, and only one steady state, corresponding to the survival of at most one species in the chemostat, is a local attractor. Numerical simulations reproduce the qualitative feature of some experimental data which show that the population density of the winning species approaches a positive steady state via transient oscillations while that of the losing species approaches the zero steady state monotonically. Received 4 August 1995; received in revised form 15 December 1995     

Dynamics and comparative statics of mutualistic communities

Within the network hypothesis proposed by Jerne, the immune response is interpreted as a collective behaviour of different antibody species, interacting through idiotypic recognition. In order to insure the stability of the network, only a few species would be implied in the response to an antigenic challenge.We study a network made up of small cycles of idiotypic units, each element activating the subsequent one and repressing the preceding one. In the recent theoretical models, the kinetics is described by steep sigmoidal functions with a repression threshold lower than the stimulation one. To enable a systematic qualitative analysis of the dynamics, we replace the continuous kinetics by stepfunctions. The antibodies are thus considered as control elements like genes, enzymes or neurones. In order to account for the different thresholds, we use discrete three-level variables.We develop two methods to study the dynamics: the first one, due to Glass, describes the time-evolution of a cycle by a system of piecewise linear (PL) differential equations and the second method is the boolean formalization, applied extensively by Thomas in the field of genetic regulation.These techniques provide complementary informations about the dynamics of the cycle: the PL method establishes a state transition diagram providing all the potential behaviours independently of the parameter values in the model, whereas the purely logical analysis permits a simulation of the trajectories for precise values of the parameters.The state transition diagram presents several steady states. It suggests to interpret the response to an antigenic challenge as a transition from one steady state to another. The multiplicity of the steady states might be associated with the various modes of immune response depending on the doses of antigen injected and on the previous antigenic history of the system.     

Non-equilibrium thermodynamics of temporally oscillating chemical reactions.

Two thermodynamic quantities are introduced: the entropy change due to a variation of chemical affinity from a steady, state. to some other state and the corresponding entropy production. The entropy change is always negative definite except at the steady state and is capable of being a Liapunov function. The phase-plane behaviour of the entropy production along the trajectory generated by kinetic equations is investigated in connection with the stability of steady state. The examples taken on that occasion are the Volterra-Lotka and Prigogine-Lefever models. The non-equilibrium thermodynamic properties common to the oscillating reactions in two-variable system are in general considered with an emphasis on the thermodynamic analysis for the direction of rotation of the trajectory generated by the two-variable kinetic equations.     

Breaking symmetry in non-planar bifurcations: distribution of flow and wall shear stress

Non-planarity in blood vessels is known to influence arterial flows and wall shear stress. To gain insight, computational fluid dynamics (CFD) has been used to investigate effects of curvature and out-of-plane geometry on the distribution of fluid flows and wall shear stresses in a hypothetical non-planar bifurcation. Three-dimensional Navier-Stokes equations for a steady state Newtonian fluid were solved numerically using a finite element method. Non-planarity in one of the two daughter vessels is found to deflect flow from the inner wall of the vessel to the outer wall and to cause changes in the distribution of wall shear stresses. Results from this study agree to experimental observations and CFD simulations in the literature, and support the view that non-planarity in blood vessels is a factor with important haemodynamic significance and may play a key role in vascular biology and pathophysiology.     

An age-structured within-host HIV-1 infection model with virus-to-cell and cell-to-cell transmissions

In this paper, a within-host HIV-1 infection model with virus-to-cell and direct cell-to-cell transmission and explicit age-since-infection structure for infected cells is investigated. It is shown that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number. By analysing the corresponding characteristic equations, the local stability of an infection-free steady state and a chronic-infection steady state of the model is established. By using the persistence theory in infinite dimensional system, the uniform persistence of the system is established when the basic reproduction number is greater than unity. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is shown that if the basic reproduction number is less than unity, the infection-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the chronic-infection steady state is globally asymptotically stable. Numerical simulations are carried out to illustrate the feasibility of the theoretical results.     

Theoretical study of flow,pressure and solute concentration in double membrane systems

A model system consisting of two rigidly held membranes in series was investigated through the application of the Kedem and Katchalsky thermodynamic single membrane flow equations. This analysis results in predictions of the steady state flow properties as well as values for the solute concentration and pressure of the internal compartment when the system is under the influence of a constant solute concentration or hydrostatic pressure gradient. It is demonstrated that although the flow properties and internal compartment pressure are complicated functions of the membrane permeability coefficients and driving gradient across the system, the relationships are greatly simplified by the explicit appearance of the internal compartment steady state solute concentration in the equations. It is shown that the steady state volume flow rate depends on the absolute value of the solute concentration in the external compartments, as well as the solute concentration gradient across the system. The properties of non-linear dependence of volume flow on concentration gradient, and rectification of volume flow are discussed and shown to be independent properties of the system. For the system under the influence of a solute concentration gradient, the internal compartment pressure can be greater or less than the ambient pressure, and depends mainly on the order in which the membranes are encountered by the volume flow. These properties are qualitatively correlated with certain available experimental observations in biological systems.     

Seed dispersal in a patchy environment with global age dependence

A model of seed population dynamics proposed by S. A. Levin, A. Hastings, and D. Cohen is presented and analyzed. With the environment considered as a mosaic of patches, patch age is used along with time as an independent variable. Local dynamics depend not only on the local state, but also on the global environment via dispersal modelled by an integral over all patch ages. Basic technical properties of the time varying solutions are examined; necessary and sufficient conditions for nontrivial steady states are given; and general sufficient conditions for global asymptotic stability of these steady states are established. Primary tools of analysis include a hybrid Picard iteration, fixed point methods, monotonicity of solution structure, and upper and lower solutions for differential equations.This work was supported in part by National Science Foundation Grants MCS-7903497 and MCS-790349701     

Analytic Approximate Solutions to the Boundary Layer Flow Equation over a Stretching Wall with Partial Slip at the Boundary

Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations.     

Global dynamics of an epidemiological model with age of infection and disease relapse

In this paper, an epidemiological model with age of infection and disease relapse is investigated. The basic reproduction number for the model is identified, and it is shown to be a sharp threshold to completely determine the global dynamics of the model. By analysing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state of the model is established. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is verified that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable, and hence the disease dies out; if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable and the disease becomes endemic.