Stochastic modeling of length-dependent telomere shortening in Corvus monedula.

It was recently shown that, within individuals, longer telomeres shorten at a higher rate. This explorative study deals with a mathematical model of this process. It is a nonlinear differential equation describing length-dependent decrease that can be linked to a Poisson process. The model also takes in account telomere shortening due to the end replication problem. Parameters are fitted using data from samples of red blood cells of free-living juvenile corvids. The Poisson process can be related to oxidative stress causing DNA strand breaks. The shortest telomeres in a genome are the best predictors of survival, and one can therefore hypothesize on functional grounds that short telomeres should be better protected by some control mechanism in the cellular system. However, the present study shows that such a mechanism is not required to explain length-dependent telomere shortening: agents of telomere shortening such as oxidative stress with a certain strength modeled by a Poisson process with an appropriately chosen parameter suffice to generate the observed pattern.

On the robustness of the geometrical model for cell wall deposition.

All plant cells are provided with the necessary rigidity to withstand the turgor by an exterior cell wall. This wall is composed of long crystalline cellulose microfibrils embedded in a matrix of other polysaccharides. The cellulose microfibrils are deposited by mobile membrane bound protein complexes in remarkably ordered lamellar textures. The mechanism by which these ordered textures arise, however, is still under debate. The geometrical model for cell wall deposition proposed by Emons and Mulder (Proc. Natl. Acad. Sci. 95, 7215-7219, 1998) provides a detailed approach to the case of cell wall deposition in non-growing cells, where there is no evidence for the direct influence of other cellular components such as microtubules. The model successfully reproduces even the so-called helicoidal wall; the most intricate texture observed. However, a number of simplifying assumptions were made in the original calculations. The present work addresses the issue of the robustness of the model to relaxation of these assumptions, by considering whether the helicoidal solutions survive when three aspects of the model are varied. These are: (i) the shape of the insertion domain, (ii) the distribution of lifetimes of individual CSCs, and (iii) fluctuations and overcrowding. Although details of the solutions do change, we find that in all cases the overall character of the helicoidal solutions is preserved.

A new class of non-linear stochastic population models with mass conservation.

We study the effects of random feeding, growing and dying in a closed nutrient-limited producer/consumer system, in which nutrient is fully conserved, not only in the mean, but, most importantly, also across random events. More specifically, we relate these random effects to the closest deterministic models, and evaluate the importance of the various times scales that are involved. These stochastic models differ from deterministic ones not only in stochasticity, but they also have more details that involve shorter times scales. We tried to separate the effects of more detail from that of stochasticity. The producers have (nutrient) reserve and (body) structure, and so a variable chemical composition. The consumers have only structure, so a constant chemical composition. The conversion efficiency from producer to consumer, therefore, varies. The consumers use reserve and structure of the producers as complementary compounds, following the rules of Dynamic Energy Budget theory. Consumers die at constant specific rate and decompose instantaneously. Stochasticity is incorporated in the behaviour of the consumers, where the switches to handling and searching, as well as dying are Poissonian point events. We show that the stochastic model has one parameter more than the deterministic formulation without time scale separation for conversions between searching and handling consumers, which itself has one parameter more than the deterministic formulation with time scale separation for these conversions. These extra parameters are the contributions of a single individual producer and consumer to their densities, and the ratio of the two, respectively. The tendency to oscillate increases with the number of parameters. The focus bifurcation point has more relevance for the asymptotic behaviour of the stochastic model than the Hopf bifurcation point, since a randomly perturbed damped oscillation exhibits a behaviour similar to that of the stochastic limit cycle particularly near this bifurcation point. For total nutrient values below the focus bifurcation point, the system gradually becomes more confined to the direct neighbourhood of the isocline for which the producers do not change.

Contest and Scramble Competition and the Carry-Over Effect in Globodera spp. in Potato-Based Crop Rotations Using an Extended Ricker Model.

The Ricker model extended with a linear term was used to model the dynamics of a potato cyst nematode population on different potato cultivars over a wide range of population densities. The model accounts for contest and scramble competition and between-year carryover of unhatched eggs. Contest competition occurs due to the restricted amount of available root sites that are the feeding source of the female nematode. Nematodes not reaching such a feeding site turn into males and do not contribute to a new generation. Scramble competition results in a decrease of the number of eggs per cyst at high densities due to the decrease in the food supply per feeding site. At still higher densities, the size of the root system declines; then dynamics are mostly governed by carryover of cysts between subsequent years. The restricted number of three parameters in the proposed model made it possible to calculate the equilibrium densities and to obtain analytical expressions of the model's sensitivity to parameter change. The population dynamics model was combined with a yield-loss assessment model and, using empirical Bayesian methods, was fitted to data from a 3-year experiment carried out in the Netherlands. The experiment was set up around the location of a primary infestation of Globodera pallida in reclaimed polder soil. Due to a wide range of population densities at short distances from the center of the infestation, optimal conditions existed for studying population response and damage in different cultivars. By using the empirical Bayesian methods it is possible to estimate all parameters of the dynamic system, in contrast to earlier studies with realistic biological models where convergence of parameter estimation algorithms was a problem. Applying the model to the outcome of the experiment, we calculated the minimum gross margin that a fourth crop needs to reach in order to be taken up in a 3-year rotation with potato. An equation was derived that accounted for both gross margin changes and nematode-related yield loss. The new model with its three parameters has the right level of complexity for the amount and type of collected data. Two other important models from the literature, containing five and 10 parameters respectively, may at this point turn out to be less appropriate. Consequences for research priorities are discussed and prediction schemes are taken in consideration.

The multifractal structure of arterial trees.

Fractal properties of arterial trees are analysed using the cascade model of turbulence theory. It is shown that the branching process leads to a non-uniform structure at the micro-level meaning that blood supply to the tissue varies in space. From the model it is concluded that, depending on the branching parameter, vessels of a specific size contribute dominantly to the blood supply of tissue. The corresponding tissue elements form a dense set in the tissue. Furthermore, if blood flow in vessels can get obstructed with some probability, the above set of tissue elements may not be dense anymore. Then there is the risk that, spread out over the tissue, nutrient and gas exchange fall short.

Mathematical conservation ecology: a one-predator-two-prey system as case study.

A method is presented to analyse the long-term stochastic dynamics of a biological population that is at risk of extinction. From the full ecosystem the method extracts the minimal information to describe the long-term dynamics of that population by a stochastic logistic system. The method is applied to a one-predator-two-prey model. The choice of this example is motivated by a study on the near-extinction of a porcupine population by mountain lions whose presence is facilitated by mule deer taking advantage of a change in land use. The risk of extinction is quantified by the expected time of extinction of the population.

A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control.

A two-component differential equation model is formulated for a host-parasitoid interaction. Transient dynamics and population crashes of this system are analysed using differential inequalities. Two different cases can be distinguished: either the intrinsic growth rate of the host population is smaller than the maximum growth rate of the parasitoid or vice versa. In the latter case, the initial ratio of parasitoids to hosts should exceed a given threshold, in order to (temporarily) halt the growth of the host population. When not only oviposition but also host-feeding occurs the dynamics do not change qualitatively. In the case that the maximum growth rate of the parasitoid population is smaller than the intrinsic growth rate of the host, a threshold still exists for the number of parasitoids in an inundative release in order to limit the growth of the host population. The size of an inundative release of parasitoids, which is necessary to keep the host population below a certain level, can be determined from the two-component model. When parameter values for hosts and parasitoids are known, an effective control of pests can be found. First it is determined whether the parasitoids are able to suppress their hosts fully. Moreover, using our simple rule of thumb it can be assessed whether suppression is also possible when the relative growth rate of the host population exceeds that of the parasitoid population. With a numerical investigation of our simple system the design of parasitoid release strategies for specific situations can be computed.

Stochastic epidemics: the expected duration of the endemic period in higher dimensional models.

A method is presented to approximate the long-term stochastic dynamics of an epidemic modelled by state variables denoting the various classes of the population such as in SIR and SEIR model. The modelling includes epidemics in populations at different locations with migration between these populations. A logistic stochastic process for the total infectious population is formulated; it fits the long-term stochastic behaviour of the total infectious population in the full model. A good approximation is obtained if only the dynamics near the equilibria is fit.

Stochastic epidemics: major outbreaks and the duration of the endemic period.

A study is made of a two-dimensional stochastic system that models the spread of an infectious disease in a population. An asymptotic expression is derived for the probability that a major outbreak of the disease will occur in case the number of infectives is small. For the case that a major outbreak has occurred, an asymptotic approximation is derived for the expected time that the disease is in the population. The analytical expressions are obtained by asymptotically solving Dirichlet problems based on the Fokker-Planck equation for the stochastic system. Results of numerical calculations for the analytical expressions are compared with simulation results.

Reconstruction of the seasonally varying contact rate for measles.

With the extended Kalman filter the time dependent contact rate for measles in the SEIR-model is reconstructed from data of the incidence of this infectious disease in the city of New York. It is concluded that although these data show through the years an irregular change in the number of infected children the contact rate is definitely periodic and follows the season. The analysis gives improved values of the parameters in the SEIR-model for this special problem.

A deterministic model of the cell cycle.

The variability of the duration of the cell cycle is explained by the phenomenon of sensitive dependence upon initial conditions; as may occur in deterministic non-linear systems. Chaotic dynamics of a system is the result of this sensitive dependence. First a deterministic system is formulated that is equivalent to the Smith-Martin transition probability model of the cell cycle. Next the model is extended to a dynamic process that ranges over the cell generations. A deterministic non-linear relationship between the cycle time of the mother and daughter cell is established. It clarifies the variability of mother-daughter correlation for the different cell types. The model is fitted to two different cell cultures; it shows that the graph of the non-linear relation has the same shape for different cell types.