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PubMed2005

Random genetic drift and gamete frequency.

An analytic expression of conditional expectation of transient gamete frequency, given that one of the two loci remains polymorphic, is obtained in terms of the diffusion process by calculating the moments of the distribution. Using this expression, a model where linkage disequilibrium is introduced by a single mutation is considered. The conditional expectation of the gamete frequency given that the locus with the mutant allele remains polymorphic is presented. The behavior is significantly different from the monotonic decrease observed in the deterministic model without random genetic drift.

Computer Simulation↗
PubMed1997

Exploring the fitness landscapes of lattice proteins.

We present methods to investigate the sequence to structure relation for proteins. We use random structures of HP-type lattice models as a coarse grained model to study generic properties of biopolymers. To circumvent the computational limitations imposed by most lattice protein folding algorithms we apply a simple and fast deterministic approximation algorithm with a tunable accuracy. We investigate ensemble properties such as the conditional probability to find structures with a certain similarity at a given distance of the underlying sequence for various alphabets. Our results suggest that the structure landscapes for lattice proteins are generally very rugged, while larger alphabets fine tune the folding process and smoothen the map. This implies a simplification for evolutionary strategies. The applied methods appear to be helpful in the study of the complex interplay between folding strategies, energy functions and alphabets. Possible implications to the investigation of evolutionary strategies or the optimization of biopolymers are discussed.

Algorithms↗
PubMed2006

Phase resetting and coupling of noisy neural oscillators.

A number of experimental groups have recently computed Phase Response Curves (PRCs) for neurons. There is a great deal of noise in the data. We apply methods from stochastic nonlinear dynamics to coupled noisy phase-resetting maps and obtain the invariant density of phase distributions. By exploiting the special structure of PRCs, we obtain some approximations for the invariant distributions. Comparisons to Monte-Carlo simulations are made. We show how phase-dependence of the noise can move the peak of the invariant density away from the peak expected from the analysis of the deterministic system and thus lead to noise-induced bifurcations.

Action Potentials↗
PubMed2003

Estimating the predicted environmental concentration of the residues of veterinary medicines: should uncertainty and variability be ignored?

European directives require that all veterinary medicines be assessed to determine the harmful effects that their use may have on the environment. Fundamental to this assessment is the calculation of the predicted environmental concentration (PEC), which is dependent on the type of drug, its associated treatment characteristics, and the route by which residues enter the environment. Deterministic models for the calculation of the PEC have previously been presented. In this article, the inclusion of variability and uncertainty within such models is introduced. In particular, models for the calculation of the PEC for residues excreted directly onto pasture by grazing animals are considered and comparison of deterministic and stochastic results suggest that uncertainty and variability cannot be ignored.

Animals↗
PubMed1976

Stochastic selection in both haplophase and diplophase.

A population genetic model incorporating the evolutionary forces of zygotic selection, gametic selection and non-Mendelian segregation has been analyzed for the case in which all selection coefficients and the segregation parameter are assumed to be random variables that are uncorrelated from generation to generation. The diffusion approximation of the model is developed, and the subsequent analysis shows that one of four limiting outcomes of the stochastic process may obtain--an allele may be fixed or lost almost surely and irrespective of the initial gene frequency, the gene frequency may converge to a unique stationary distribution, or an allele may be fixed or lost with probabilities depending on the initial gene frequency. These outcomes correspond rather closely with the possible outcomes of the deterministic model--fixation or loss of an allele, convergence to a stable equilibrium, or the existence of an unstable equilibrium.

Alleles↗
PubMed2003

Response of a deterministic epidemiological system to a stochastically varying environment.

Fluctuations in the natural environment introduce variability into the biological systems that exist within them. In this paper, we develop a model for the influence of random fluctuations in the environment on a simple epidemiological system. The model describes the infection of a dynamic host population by an environmentally sensitive pathogen and is based on the infection of sugar beet plants by the endoparasitic slime-mold vector Polymyxa betae. The infection process is switched on only when the temperature is above a critical value. We discuss some of the problems inherent in modeling such a system and analyze the resulting model by using asymptotic techniques to generate closed-form solutions for the mean and variance of the net amount of new inoculum produced within a season. In this way, the variance of temperature profile can be linked with that of the inoculum produced in a season and hence the risk of disease. We also examine the connection between the model developed in this paper and discrete Markov-chain models for weather.

Beta vulgaris↗
PubMed1991

Fractal geometry: a design principle for living organisms.

Fractal geometry allows structures to be quantitatively characterized in geometric terms even if their form is not even or regular, because fractal geometry deals with the geometry of hierarchies and random processes. The hypothesis is explored that fractal geometry serves as a design principle in biological organisms. The internal membrane surface of cells, or the inner lung surface, are difficult to describe in terms of classical geometry, but they are found to show properties describable by fractal geometry, at least sectionwise and within certain bounds set by deterministic design properties. Concepts of fractal geometry are most useful in characterizing the structure of branching trees, such as those found in pulmonary airways and in blood vessels. This explains how the large internal gas exchange surface of the lung can be homogeneously and efficiently ventilated and perfused at low energetic cost. It is concluded that to consider fractal geometry as a biological design principle is heuristically most productive and provides insights into possibilities of efficient genetic programming of biological form.

Animals↗
PubMed1996

Estimating confidence intervals for cost-effectiveness ratios: an example from a randomized trial.

Cost-effectiveness ratios usually appear as point estimates without confidence intervals, since the numerator and denominator are both stochastic and one cannot estimate the variance of the estimator exactly. The recent literature, however, stresses the importance of presenting confidence intervals for cost-effectiveness ratios in the analysis of health care programmes. This paper compares the use of several methods to obtain confidence intervals for the cost-effectiveness of a randomized intervention to increase the use of Medicaid's Early and Periodic Screening, Diagnosis and Treatment (EPSDT) programme. Comparisons of the intervals show that methods that account for skewness in the distribution of the ratio estimator may be substantially preferable in practice to methods that assume the cost-effectiveness ratio estimator is normally distributed. We show that non-parametric bootstrap methods that are mathematically less complex but computationally more rigorous result in confidence intervals that are similar to the intervals from a parametric method that adjusts for skewness in the distribution of the ratio. The analyses also show that the modest sample sizes needed to detect statistically significant effects in a randomized trial may result in confidence intervals for estimates of cost-effectiveness that are much wider than the boundaries obtained from deterministic sensitivity analyses.

Analysis of Variance↗
PubMedDate not supplied

Evolution and persistence of influenza A and other diseases.

The evolution of the etiological agents of disease presents one of the greatest challenges for their control, and makes essential complementing standard epidemiological investigations with broader approaches that allow for evolutionary change. Given the stunning genetic diversity that is possible for many such agents, such as the influenza virus, it is impossible to represent all of the diversity manifest at the level of amino acid sequences. We show that drift-variant influenza strains naturally cluster into groups which are associated with functionally important epitopic regions. Dominant clusters typically replace each other every 2-5 years, and this feature is fundamental to the development of vaccination strategies. We furthermore show that stochastic fluctuations can greatly magnify small interference effects among strains, or even among subtypes, leading for example to competitive exclusion in situations where such effects would be unexpected based on the usual deterministic models. We suggest that this effect might be involved in the explanations of some persistent empirical anomalies.

Animals↗
PubMed2004

Stochastic simulation of the mammalian circadian clock.

Circadian (nearly 24-h) clocks are remarkably accurate at timing biological events despite the randomness of their biochemical reactions. Here we examine the causes of their immunity to molecular noise in the context of a detailed stochastic mathematical model of the mammalian circadian clock. This stochastic model is a direct generalization of the deterministic mammalian circadian clock model previously developed. A feature of that model is that it completely specifies all molecular reactions, leaving no ambiguity in the formulation of a stochastic version of the model. With parameters based on experimental data concerning clock protein concentrations within a cell, we find accurate circadian rhythms in our model only when promoter interaction occurs on the time scale of seconds. As the model is scaled up by proportionally increasing the numbers of molecules of all species and the reaction rates with the promoter, the observed variability scales as 1/n(0.5), where n is the number of molecules of any species. Our results show that gene duplication increases robustness by providing more promoters with which the transcription factors of the model can interact. Although PER2 mutants were not rhythmic in the deterministic version of this model, they are rhythmic in the stochastic version.

Animals↗
PubMed2003

Reproductive uncertainty and the relative competitiveness of simultaneous hermaphroditism versus dioecy.

Hermaphroditism is typically associated with a sedentary existence, whereas dioecy is associated with mobility. This pattern is reflected within flowering plants, as dioecious species commonly possess traits that promote high dispersal. We investigated these associations with three population dynamics models (an individual-based simulation and two mathematical models, one deterministic and the other stochastic) that allowed us to examine competition for space between a hermaphroditic and dioecious species from different perspectives. The competing species are identical in every way but their sexual system. Separation of the sexes increases the variances of pollen import and seed dispersal for the dioecious species. These variances propagate through subsequent reproductive processes and ultimately reduce mean recruitment as a result of nonlinear averaging (Jensen's inequality). A dioecious species could overcome this disadvantage simply by producing more gametes than hermaphrodites; however, in line with the association with mobility, selection on dioecious species should also favor traits that reduce reproductive uncertainty, such as extensive dispersal.

Biological Evolution↗
PubMed2001

Real-Time linux dynamic clamp: a fast and flexible way to construct virtual ion channels in living cells.

We describe a system for real-time control of biological and other experiments. This device, based around the Real-Time Linux operating system, was tested specifically in the context of dynamic clamping, a demanding real-time task in which a computational system mimics the effects of nonlinear membrane conductances in living cells. The system is fast enough to represent dozens of nonlinear conductances in real time at clock rates well above 10 kHz. Conductances can be represented in deterministic form, or more accurately as discrete collections of stochastically gating ion channels. Tests were performed using a variety of complex models of nonlinear membrane mechanisms in excitable cells, including simulations of spatially extended excitable structures, and multiple interacting cells. Only in extreme cases does the computational load interfere with high-speed "hard" real-time processing (i.e., real-time processing that never falters). Freely available on the worldwide web, this experimental control system combines good performance. immense flexibility, low cost, and reasonable ease of use. It is easily adapted to any task involving real-time control, and excels in particular for applications requiring complex control algorithms that must operate at speeds over 1 kHz.

Computer Simulation↗
PubMed1989

Impact of diffuse nitrate pollution sources on groundwater quality--some examples from Czechoslovakia.

In several regions of Czechoslovakia with intensive agricultural production, the correlation between the amount of nitrogen fertilizer applied and the nitrate content in groundwater has been recognized. Nitrate pollution of groundwater is considered to be the most serious source of nonpoint pollution in Czechoslovakia. A program of research into the effects of farming activities on groundwater quality in Czechoslovakia is under way on experimental fields (20 to 30 hectares) and, simultaneously, in regions in which shallow, vulnerable aquifers occur. The importance of the soil organic matter's stability for maintaining the groundwater quality is emphasized. Research based on nitrogen and organic carbon balance has shown that the restoration of a soil-groundwater system is a complicated process that usually requires changes in the extent and intensity of agricultural activities and consistent attention to the effects produced by natural conditions. Regional investigation of the impact of farming on shallow aquifers in the fluvial deposits of the Elbe River in Bohemia has proved the hydrochemical instability and vertical hydrochemical heterogeneity of these aquifers. The WASTEN deterministic model was used for modeling the transport and transformation of various types of inorganic fertilizers. The input data is based on laboratory and field measurements. Special topics are the verification of model calculations and the time and spatial variability of input data with respect to the unsaturated zone. The research results are being used for making regional and national agro-groundwater managerial schemes more precise, as well as for decision-making.

Agriculture↗
PubMed1995

Breast implants and breast cancer--reanalysis of a linkage study.

BACKGROUND: In 1992, Berkel and colleagues reported in the Journal the results of their study of the potential association of breast augmentation and breast cancer. The study reported that women who had breast augmentation had a significantly lower subsequent risk of breast cancer (P < 0.01) than the general population, with a standardized incidence ratio of 0.48 overall. Assuming a 10-year induction period (that is, assuming that cancers found within 10 years of the augmentation might have been the result of a process begun before surgery and therefore should not be considered), the reported standardized incidence ratio was 0.16. Problems were later identified involving some of the study methods. This paper reports a second analysis of these data. METHODS: We used a data set from Alberta Health Care to identify eligible women with bilateral breast augmentation. Using a combination of deterministic and probabilistic methods, we linked this data set to the Alberta Cancer Registry to identify subsequent breast cancers that developed during the study period. Multiple estimates of standardized incidence ratios were calculated on the basis of differing study-eligibility dates, induction periods, and types of breast-cancer (invasive only or invasive plus in situ). RESULTS: The reanalysis found substantial differences in the numbers of person-years at risk, resulting in higher standardized incidence ratios than in the original analysis. The final ratios for all breast cancers, with October 1, 1973, used as the starting date of the study, were 0.76 (95 percent confidence interval, 0.55 to 1.02), 0.85 (95 percent confidence interval, 0.58 to 1.19), and 0.68 (95 percent confidence interval, 0.32 to 1.25) for induction periods of 0, 5, and 10 years, respectively. None of these standardized incidence ratios were significantly different from 1. CONCLUSIONS: On the basis of this reanalysis, the incidence of breast cancer among the women who had breast augmentation could not be said to be either significantly higher or lower than that among the general population over the period during which this cohort was followed.

Adult↗
PubMed2005

Normal tissue radiosensitivity: prediction on deterministic or stochastic basis?

For the same standardized physical radiation dose there is considerable variation, among different patients, of the magnitude of early and late normal tissue reactions. Technical and clinical factors account for about one third only of these variations. Genetic or epigenetic differences between patients account for the greater proportion of interpatient radiosensitivity differences. Attempts have been made to correlate differences in the tissue radiosensitivity with the in vitro radiosensitivity of fibroblasts or lymphocytes as well as with other biological cellular processes related to cell death (mainly DNA repair and chromosomal aberrations). Apart from some genetic diseases such as ataxia telangiectasia, there were positive results in some studies that could not be reproduced in others. It is now realized that for normal tissue radiation-induced morbidity (a) cell kill is not the only factor, (b) interaction with a large number of gene products, such as IL-2, IL-6, TGF-beta, is involved, and (c) radiosensitivity differences can reflect genetic differences. Polymorphism in a wide range of genes can now be demonstrated and analyzed as a source of variations in the radiation response. cDNA microarray procedures can also allow simultaneous measurement, in the same sample, of thousands of genes and thus avoiding basing the investigation on a restricted number of genes. Preliminary results could show retrospectively a correlation between late normal tissue reactions and expression of certain genes. Using cDNA microarray before radiotherapy to predict the occurrence of serious late reactions is still under consideration. Theoretically, an efficient predictor procedure may serve in (a) detection of hypersensitive patients before submitting them to highly toxic radiotherapy regimes such as whole body irradiation, (b) detection of hypersensitive patients in clinical situations where radiotherapy can be effectively replaced by other modalities, and (c) selection of patients for elaborate radiotherapy techniques involving dose escalation to high levels.

Dose Fractionation, Radiation↗
PubMed2004

Noise-induced phase locking in coupled coherence-resonance oscillators.

We investigate the effects of additive noise on coupled excitable chemical oscillators, particularly focusing on how oscillatory coupled modes can be induced by noise. We find that phase locking in the weak coupling regime occurs through coherence resonance, although the resulting phase locking modes are apparently similar to those in coupled deterministic oscillators. Experimental observations are approximately reproduced in a numerical simulation with a forced Oregonator reaction-diffusion model.

Adaptation, Physiological↗
PubMed2007

A stochastic model for circadian rhythms from coupled ultradian oscillators.

BACKGROUND: Circadian rhythms with varying components exist in organisms ranging from humans to cyanobacteria. A simple evolutionarily plausible mechanism for the origin of such a variety of circadian oscillators, proposed in earlier work, involves the non-disruptive coupling of pre-existing ultradian transcriptional-translational oscillators (TTOs), producing "beats," in individual cells. However, like other TTO models of circadian rhythms, it is important to establish that the inherent stochasticity of the protein binding and unbinding does not invalidate the finding of clear oscillations with circadian period. RESULTS: The TTOs of our model are described in two versions: 1) a version in which the activation or inhibition of genes is regulated stochastically, where the 'unoccupied" (or "free") time of the site under consideration depends on the concentration of a protein complex produced by another site, and 2) a deterministic, "time-averaged" version in which the switching between the "free" and "occupied" states of the sites occurs so rapidly that the stochastic effects average out. The second case is proved to emerge from the first in a mathematically rigorous way. Numerical results for both scenarios are presented and compared. CONCLUSION: Our model proves to be robust to the stochasticity of protein binding/unbinding at experimentally determined rates and even at rates several orders of magnitude slower. We have not only confirmed this by numerical simulation, but have shown in a mathematically rigorous way that the time-averaged deterministic system is indeed the fast-binding-rate limit of the full stochastic model.

Animals↗
PubMed2006

E. coli microcosms indicate a tight link between predictability of ecosystem dynamics and diversity.

The diversity-stability hypothesis proposes that ecosystem diversity is positively correlated with stability. The impact of ecosystem diversity is, however, still debated. In a microcosm experiment using diverged Escherichia coli cells, we show that the fitness of community members depends on the complexity (number of participants) of the system. Interestingly, the spread of a community member with a superior genotype is mostly stochastic in low-complexity systems, but highly deterministic in a more complex environment. We conclude that system complexity provides a buffer against stochastic effects.

Alleles↗