Stochastic epidemics in dynamic populations: quasi-stationarity and extinction

Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for τ, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how τ depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours. Received: 14 February 2000 / Revised version: 5 June 2000 / Published online: 24 November 2000     

Mathematical models and simulations of bacterial growth and chemotaxis in a diffusion gradient chamber

The diffusion gradient chamber (DGC) is a novel device developed to study the response of chemotactic bacteria to combinations of nutrients and attractants [7]. Its purpose is to characterize genetic variants that occur in many biological experiments. In this paper, a mathematical model which describes the spatial distribution of a bacterial population within the DGC is developed. Mathematical analysis of the model concerning positivity and boundedness of the solutions are given. An ADI (Alternating Direction Implicit) method is constructed for finding numerical solutions of the model and carrying out computer simulations. The numerical results of the model successfully reproduced the patterns that were observed in the experiments using the DGC. Received: 3 June 1997 / Revised version: 15 August 2000 / Published online: 20 December 2000     

An alternative stochastic model of generation of oligodendrocytes in cell culture

According to our previous model, oligodendrocyte – type 2 (O-2A) astrocyte progenitor cells become competent for differentiation in vitro after they complete a certain number of critical mitotic cycles. After attaining the competency to differentiate, progenitor cells divide with fixed probability p in subsequent cycles. The number of critical cycles is random; analysis of data suggests that it varies from zero to two. The present paper presents an alternative model in which there are no critical cycles, and the probability that a progenitor cell will divide again decreases gradually to a plateau value as the number of completed mitotic cycles increases. In particular all progenitor cells have the ability to differentiate from the time of plating. The Kiefer-Wolfowitz procedure is used to fit the new model to experimental data on the clonal growth of purified O-2A progenitor cells obtained from the optic nerves of 7 day old rats. The new model is shown to fit the experimental data well, indicating that it is not possible to determine whether critical cycles exist on the basis of these experimental data. In contrast to the fit of the previous model, which suggested that the addition of thyroid hormone increased the limiting probability of differentiation as the number of mitotic cycles increases, the fit of the new model suggests that the addition of thyroid hormone has almost no effect on the limiting probability of differentiation. Received: 6 March 2000 / Revised version: 18 September 2000 / Published online: 30 April 2001     

Local genetic differentiation in Proteopsis argentea (Asteraceae), a perennial herb endemic in Brazil

 Proteopsis argentea (Asteraceae, Vernonieae) is a perennial herb endemic to the “campos rupestres” of the Espinha?o Range in Minas Gerais, Brazil, with fragmented populational distribution. Eleven populations were sampled, throughout the entire distribution of the species, and assayed for isozyme variation. Low intra-population genetic diversity was found (P = 19.2; A = 1.30; He = 0.058) whereas species level diversity was higher (P = 55.5, A = 2.0, He = 0.093). The most geographically isolated population showed exclusive alleles at two loci, whereas two populations less than 2 km apart from each other showed inverted frequencies for two alleles. Mean genetic identity was high (I = 0.974), but the large Fst (0.30) indicates that the species could lose an important part of its genetic variation with the extinction of a single population. Our findings indicate that geographic isolation alone cannot explain population differentiation: localized pollinator behaviour and selection, for example, may be contributing to the patterns observed. Received February 18, 2000 Accepted November 1, 2000     

Transient dynamics and early diagnostics in infectious disease

To date, mathematical models of the dynamics of infectious disease have consistently focused on understanding the long-term behavior of the interacting components, where the steady state solutions are paramount. However for most acute infections, the long-term behavior of the pathogen population is of little importance to the host and population health. We introduce the notion of transient pathology, where the short-term dynamics of interaction between the immune system and pathogens is the principal focus. We identify the amplifying effect of the absence of a fully operative immune system on the pathogenesis of the initial inoculum, and its implication for the acute severity of the infection. We then formalize the underlying dynamics, and derive two measures of transient pathogenicity: the peak of infection (maximum pathogenic load) and the time to peak of infection, both crucial to understanding the early dynamics of infection and its consequences for early intervention. Received: 25 January 2000 / Revised version: 30 November 2000 / Published online: 12 October 2001     

Enzymology of Phanerochaete chrysosporium with respect to the degradation of recalcitrant compounds and xenobiotics

The archetypal white-rot fungus Phanerochaete chrysosporium has been shown to degrade a variety of persistent environmental pollutants. Many of the enzymes responsible for pollutant degradation, which are normally involved in the degradation of wood, are extracellular. Thus, P. chrysosporium is able to degrade toxic or insoluble chemicals more efficiently than other microorganisms. P. chrysosporium has a range of oxidative and reductive mechanisms and uses highly reactive, nonspecific redox mediators which increase the number of chemicals that can be effectively degraded. This review gives an overview of the enzymes that are believed to be important for bioremediation and briefly discusses the degradation of some individual chemicals. Received: 25 April 2000 / Received revision: 05 June 2000 / Accepted: 04 July 2000     

Dynamics of adhesive rupture between fibroblasts and fibronectin: microplate manipulations and deterministic model

To characterize the dynamics of cell-substrate adhesive rupture, we used a novel micromanipulation technique, in which individual fibroblasts seized on a rigid microplate were placed into contact with a fibronectin-coated flexible microplate, then pulled away. The fibronectin density (0–3000 molecules/μm²) and the pulling rate (1–10 μm/s) were controlled. The extent of the contact zone decreased to zero at a time threshold corresponding to adhesive rupture. The uniaxial force at the interface, computed from the deflection of the microplate, increased linearly with time and reached a maximum before dropping to zero. A deterministic model, focusing on the mean number of bonds between fibronectin and its membrane receptor on the cell surface, shows rapid rupture when the force reaches a critical value, in agreement with experimental observations. Increasing the ligand density and the rate of load raises the maximal force (30–200 nN), in reasonably good agreement with the model predictions. Minimization of error between experimental and simulated forces allowed identification of two physicochemical properties of the bond, i.e. its association rate constant (k ^2D _on=3 × 10⁻⁴ μm²/s) and structural length (d=3 nm). These results may help understand better fibroblast locomotion and interaction with the extracellular matrix. Received: 2 November 1999 / Revised version: 23 March 2000 / Accepted: 19 April 2000     

Short-term binding of fibroblasts to fibronectin: optical tweezers experiments and probabilistic analysis

The biophysical properties of the interaction between fibronectin and its membrane receptor were inferred from adhesion tests on living cells. Individual fibroblasts were maintained on fibronectin-coated glass for short time periods (1–16 s) using optical tweezers. After contact, the trap was removed quickly, leading to either adhesion or detachment of the fibroblast. Through a stochastic analysis of bond kinetics, we derived equations of adhesion probability versus time, which fit the experimental data well and were used to compute association and dissociation rates (k ₊=0.3–1.4 s⁻¹ and k _off=0.05–0.25 s⁻¹, respectively). The bond distribution is binomial, with an average bond number ≤10 at these time scales. Increasing the fibronectin density (100–3000 molecules/μm²) raised k ₊ in a diffusion-dependent manner, leaving k _off relatively unchanged. Increasing the temperature (23–37 °C) raised both k ₊ and k _off, allowing calculation of the activation energy of the chemical reaction (around 20 k _B T). Increasing the compressive force on the cell during contact (up to 60 pN) raised k ₊ in a logarithmic manner, probably through an increase in the contact area, whereas k _off was unaffected. Finally, by varying the pulling force to detach the cell, we could distinguish between two adhesive regimes, one corresponding to one bond, the other to at least two bonds. This transition occurred at a force around 20 pN, interpreted as the strength of a single bond. Received: 2 November 1999 / Revised version: 6 March 2000 / Accepted: 19 April 2000     

Influence of the aeration rate on the yields of the biocontrol nematode Heterorhabditis megidis in monoxenic liquid cultures

The entomopathogenic nematode–bacterium complex Heterorhabditis megidis–Photorhabdus luminescens was cultured in 10-l internal loop bioreactors with marine impellers at aeration rates of 0.3 vvm and 0.7 vvm. Process parameters like impeller velocity and oxygen saturation were controlled at equal set points. The bacterial density was assessed at 24 h. Nematode dauer juveniles (DJ) were then inoculated and the development to adults after 8 days and final DJ yields after 16 days were recorded. The bacterial population density and the nematode inoculum development was variable and was not influenced by the aeration rate. A significant effect on the yield was recorded at the highest aeration rate. This result was confirmed by a direct comparison in two 5-l internal loop glass bioreactors at 0.3 vvm and 1.0 vvm, which were inoculated with nematode and bacterium pre-cultures from the same flask culture. Possible reasons for the positive correlation between aeration rate and DJ yield are discussed. Received: 27 September 1999 / Received revision: 21 January 2000 / Accepted: 23 January 2000     

Growth of Photorhabdus luminescens in batch and glucose fed-batch culture

Photorhabdus luminescens, a bacterial symbiont of entomopathogenic biocontrol nematodes, was grown in batch and glucose fed-batch culture. The cell density, bioluminescence, production of antibiotic substances, number of cells with inclusion bodies, glucose concentration and oxygen uptake rate were recorded. The addition of 12.4 g l⁻¹ glucose prolonged the growth, and the yield almost doubled, from 6.85 g l⁻¹ to 12.45 g l⁻¹ dry mass. The production of antibiotic substances increased by 140%. Bioluminescence was higher in the batch culture. A shift of P. luminescens to phase II variants was not detected. Received: 21 January 2000 / Received revision: 3 April 2000 / Accepted: 7 April 2000     

Heterologous protein production in methylotrophic yeasts

The facultative methylotrophic yeasts Candida boidinii, Pichia methanolica, Pichia pastoris and Hansenula polymorpha have been developed as systems for heterologous gene expression. They are based on strong and regulatable promoters for expression control derived from methanol metabolism pathway genes. An increasing number of biotechnological applications attest to their status as preferred options among the various gene expression hosts. The well-established P. pastoris and H. polymorpha systems have been utilized in especially competitive and consistent industrial-scale production processes. Pharmaceuticals and technical enzymes produced in these methylotrophs have either already entered the market or are expected to do so in the near future. The article describes the present status of the methylotrophic yeasts as expression systems, focusing on applied examples of the recent past. Received: 9 May 2000 / Received revision: 20 June 2000 / Accepted: 23 June 2000     

Hydrodynamic aspects and Arthrospira growth in two outdoor tubular undulating row photobioreactors

Two tubular undulating row photobioreactors (TURPs) with a very high illuminated surface/volume ratio (400 m⁻¹) were designed and constructed for the growth of photosynthetic micro-organisms. Experiments were conducted under outdoor conditions; and Arthrospira recycling was performed with airlifts (one for each row). The rows in each reactor faced east-west and consisted of a flexible polyvinyl chloride pipe (22 m long, 0.01 m bore) arranged in a sinusoidal shape. We studied the hydraulic performance of the sine-shaped photobioreactor rows during culture recycling in the TURPs at a very high Reynolds number (4200), when Arthrospira showed Newtonian fluid behavior. The sinusoidal pipe arrangement imposed a sine waveform on the culture, which led to better light utilization. During summer, a volumetric productivity of 2.2 g l⁻¹ day⁻¹ was reached in the TURP-5r (5 rows m⁻²), whereas an area productivity of 35 g m⁻² day⁻¹ was obtained in the TURP-10r (10 rows m⁻²). This was due to more light being available in the TURP-5r, because its rows were more spaced out and the photic ratio (R _f) was low (3.0). In the TURP-10r, the closer rows caused a dilution of the sunlight, but gave a better light distribution inside the Arthrospira culture and improved the light utilization. This was attributed to the high R _f (6.0) of this reactor. Received: 8 October 1999 / Received revision: 20 January 2000 / Accepted: 23 January 2000     

Continuous marennin production by agar-entrapped Haslea ostrearia using a tubular photobioreactor with internal illumination

The marine diatom Haslea ostrearia was immobilized in a tubular agar gel layer introduced into a photobioreactor of original design with internal illumination for the continuous synthesis of marennin, a blue-green pigment of biotechnological interest. Marennin was produced for a long-term period (27–43 days) and the volumetric productivity was maximum (18.7 mg day⁻¹ l⁻¹ gel) at the highest dilution rate (0.25 day⁻¹) and lowest agar layer thickness (3 mm). Heterogeneous cell distribution in the agar layer revealed diffusional limitation of light and nutrients. However, the 3 mm gel thickness led to a more homogeneous cell distribution during incubation and to an increase of the whole biomass in the agar gel layer. Received: 22 October 1999 / Received revision: 14 February 2000 / Accepted: 18 February 2000     

A mathematical analysis for the Brownian dynamics of a DNA tether

In the single-particle tracking experiment, the internal motion of a single DNA or polymer molecule whose one end is attached to a microsphere (optical marker) and the other end is anchored to a substratum is studied (Finzi and Gelles, 1995). The stochastic Brownian dynamics of the sphere reflect the spontaneous fluctuations, thus the physical characteristics, of the DNA or polymer molecule (Qian and Elson, 1999, Qian, 2000). In this paper, two continuous models of polymer molecules, a flexible elastic string and a weakly bentable elastic rod, are analyzed. Both models are cast mathematically in terms of linear stochastic differential equations. Based on Fourier analyses, we calculate the mean square displacement (MSD) of the particle motion, the key observable in the experiment. We obtain for both models the short-time asymptotics for the MSD, as well as the long-time behavior in terms of the smallest non-zero eigenvalues. It is shown that: (i) the long-time dynamics of continuous elastic string model quantitatively agree with that of the discrete bead-spring model. (ii) The short-time MSD of both models are controlled by the tethered particle, with linear dependence on t. (iii) The two models show characteristic difference for long-time behavior: The longest relaxation time is proportional to L ² for long elastic string and to L for short elastic string, but is proportional to L ⁴ for both long and short weakly bentable rod. Received: 26 March 1998 / Revised version: 9 June 2000 / Published online: 14 September 2000     

Chromosome numbers of the Australian Zosteraceae

 Somatic chromosome numbers of 2n = 24 are reported for all three species of Australian Zostera: Z. capricorni Aschers., Z. muelleri Irmisch ex Aschers. and Z. mucronata den Hartog and 2n = 36 for Heterozostera tasmanica (Martens ex Aschers.) den Hartog. All Australian zosteroidean species apparently have similar chromosome morphology: dot-like or rod shaped. It is suggested that the chromosome number and its morphology can be used to distinguish genera and subgenera in the Zosteraceae but not for species identification, and that speciation is not accompanied by changes of chromosome numbers. Received December 1, 1999 Accepted September 6, 2000     

Mitochondrial DNA reveals low population differentiation in elongate loach, <Emphasis Type="Italic">Leptobotia elongata</Emphasis> (Bleeker): implications for conservation

Elongate loach (Leptobotia elongata (Bleeker)), an endemic fish species to China, is a famous ornamental freshwater fish. Here, a comparative study of mtDNA control region (D-loop) (835 bp) sequences was performed to analyze its wild population structure and evaluate the genetic diversity for 110 individuals from five locations in the upper reaches of the Yangtze River, China. A total of 49 polymorphic sites and 45 haplotypes yielded high haplotype diversity (h = 0.952), but low nucleotide diversity (π = 0.00454) as that of many fish species. Sequence divergences between haplotypes ranged from 0.0033 ± 0.0011 to 0.0050 ± 0.0012 in intra-groups, and from 0.0037 ± 0.0.0011 to 0.0050 ± 0.0012 between groups. Significant values of Tajima’s D (−1.86383, P < 0.01) and Fu’s F _S (−25.93, P < 0.01), together with uni-modal mismatch distribution, indicated a recent genetic bottleneck or population expansion of the species. Analysis of molecular variance (AMOVA) indicated a small amount of differentiation among groups (1.7%); most of the total variation occurred within groups (98.3%). Also, there was no significant population structure (F _ST = 0.017, P > 0.05), and estimates of gene flows among groups were extremely high (Nm = 28.88), suggesting low genetic divergence between populations in the species. The lack of genetic differentiation among groups is most likely due to the combined gene flow from the downstream movement of eggs and larvae with currents and the upstream or downstream migration of adults throughout the distribution. These groups of L. elongata distributed in upper reaches of the Yangtze River should be considered as a single management unit.     

Changes in the physiological and agricultural characteristics of peat-based Bradyrhizobium japonicum inoculants after long-term storage

Commercial soybean inoculants processed with sterilised peat and stored at 20 °C for 1–8 years were used as experimental materials to assess the changes in the physiological activity of Bradyrhizobium japonicum after storage. Viable counts decreased and physiological characteristics of the bacterium changed during storage, with an increase in the time taken for colony appearance on a medium without yeast extract, an increase in the lag time for nodule appearance on soybean grown in glass tubes and a decrease in survival on seeds. All the inoculants produced a significant increase in grain yield in a field experiment. The percentage of efficient cells in the field (relative to the plate counts) decreased as the length of storage increased. These results suggest that the physiological activity of B. japonicum cells changes after storage. Practical implications for inoculant quality control are discussed. Received: 20 September 1999 / Received revision: 3 March 2000 / Accepted: 6 March 2000     

Landlocked Arctic charr (Salvelinus alpinus) population structure and lake morphometry in Greenland – is there a connection?

Landlocked Arctic charr (Salvelinus alpinus) populations in sub-Arctic and Arctic Greenland lakes were sampled with multi-mesh-sized survey gillnets. The study covered a range of small shallow lakes (0.01 km², maximum depth <3.3 m) to large deep lakes (43 km², maximum depth >200 m). Arctic charr were found in one to three different forms in lakes with maximum depths >3 m. A dwarf form occurred in all lakes inhabited by Arctic charr and was the only form in lakes with maximum depths <8 m. In deeper lakes with maximum depths >20 m and a surface area <0.5 km², larger charr were found, although in low numbers, the length-frequency distribution being unimodal with a tail towards large sizes. In lakes with a maximum depth >20 m, large-sized charr were more abundant, and the length-frequency distribution of the population was bimodal, with a first mode around 10–12 cm and a second mode around 26–37 cm. In a single large and deep lake, a distinct medium-sized pelagic zooplankton-eating charr form occurred. Maximum size of individual charr was significantly positively correlated with lake maximum depth and volume, and the mean size of large-sized charr was significantly positively correlated with lake volume. Our study indicates that the charr population structure became more complex with increasing lake size. Moreover, the population structure seemed to be influenced by lake-water transparency and the presence or absence of three-spined stickleback (Gasterosteus aculeatus). Accepted: 31 January 2000     

Life and death near a windy oasis

We propose a simple experiment to study delocalization and extinction in inhomogeneous biological systems. The nonlinear steady state for, say, a bacteria colony living on and near a patch of nutrient or favorable illumination (“oasis”) in the presence of a drift term (“wind”) is computed. The bacteria, described by a simple generalization of the Fisher equation, diffuse, divide A→A + A, die A→ 0, and annihilate A + A→ 0. At high wind velocities all bacteria are blown into an unfavorable region (“desert”), and the colony dies out. At low velocity a steady state concentration survives near the oasis. In between these two regimes there is a critical velocity at which bacteria first survive. If the “desert” supports a small nonzero population, this extinction transition is replaced by a delocalization transition with increasing velocity. Predictions for the behavior as a function of wind velocity are made for one and two dimensions. Received: 3 August 1998 / Revised version: 17 July 1999 / Published online: 4 July 2000     

Continuous production of enantiopure 1,2-epoxyhexane by yeast epoxide hydrolase in a two-phase membrane bioreactor

A two-phase membrane bioreactor was developed to continuously produce enantiopure epoxides using the epoxide hydrolase activity of Rhodotorula glutinis. An aqueous/organic cascade, hydrophilic, hollow-fiber membrane bioreactor was used: (1) to carry out large-scale resolution of epoxides, (2) to continuously extract residual enantiopure epoxides from the aqueous phase, and (3) to separate inhibitory formed diol from the yeast cells contained in the aqueous phase. Dodecane was employed to dissolve-feed epoxide as well as to extract residual epoxide. 1,2-Epoxyhexane was used as a model substrate. By use of this membrane bioreactor, enantiopure (S)-1,2-epoxyhexane (>98% enantiomeric excess) was obtained with a volumetric productivity of 3.8 g l⁻¹ h⁻¹. The continuous-production system was operated for 12 days and resulted in 38 g enantiopure (S)-1,2-epoxyhexane. Received: 14 February 2000 / Received revision: 15 June 2000 / Accepted: 18 June 2000