The mathematical modelling of population dynamics taking into account the adaptive behavior of individuals

The general approach for modelling of abundance dynamic of biological populations and communities is offered. The mechanisms of individual adaptation in changing environment are considered. The approach is detailed for population models without structure and with age structure. The property of solutions are investigated. As examples the author studies the concrete definitions of general models by analogy with models of Ricker and May. Theoretical analysis and calculations shows that survival of model population in extreme situation increases if adaptive behaviour is taking into account.     

A mathematical model for sarcomere mechanics in cross-striated muscles taking into account the stretch and twist of actin filaments

A model of sarcomere mechanics, which takes into account the elongation of the actin and myosin filaments and twisting of the actin filaments during muscle contraction is suggested. The model accounts for the experimentally observed phenomena of the stretch and twist of actin filaments due to strong binding of myosin heads and pulling force. Some model parameters were estimated from published experimental data. The results of modelling suggest that the twist of actin filaments may play an essential role in mechanical responses of contracting muscle fibres to stepwise changes in their length.     

Parametric and semiparametric model-based estimates of the finite population mean for two-stage cluster samples with item nonresponse

This article concerns item nonresponse adjustment for two-stage cluster samples. Specifically, we focus on two types of nonignorable nonresponse: nonresponse depending on covariates and underlying cluster characteristics, and depending on covariates and the missing outcome. In these circumstances, standard weighting and imputation adjustments are liable to be biased. To obtain consistent estimates, we extend the standard random-effects model by modeling these two types of missing data mechanism. We also propose semiparametric approaches based on fitting a spline on the propensity score, to weaken assumptions about the relationship between the outcome and covariates. These new methods are compared with existing approaches by simulation. The National Health and Nutrition Examination Survey data are used to illustrate these approaches.     

Molecular-modeling calculations of enzymatic enantioselectivity taking hydration into account

A new molecular-modeling methodology has been applied to explain enzymatic enantioselectivity in water. This methodology, which combines vacuum molecular mechanics and the continuum solvation method, should provide a more realistic view of the solvent-enzyme and solvent-substrate interactions than the heretofore used approaches involving the vacuum molecular mechanics only. The methodology described herein has been validated using the experimental data on alpha-chymotrypsin's enantioselectivity in the hydrolysis of four chiral substrates. The reasons why the vacuum molecular mechanics, although not taking hydration into account, still in most cases provide a satisfactory approximation of reality are discussed.     

Correlation between genetic diversity and environmental suitability: taking uncertainty from ecological niche models into account

The hindcast of shifts in the geographical ranges of species as estimated by ecological niche modelling (ENM) has been coupled with phylogeographical patterns, allowing the inference of past processes that drove population differentiation and genetic variability. However, more recently, some studies have suggested that maps of environmental suitability estimated by ENM may be correlated to species' abundance, raising the possibility of using environmental suitability to infer processes related to population demographic dynamics and genetic variability. In both cases, one of the main problems is that there is a wide variation in ENM development methods and climatic models. In this study, we analyse the relationship between heterozygosity (He) and environmental suitability from multiple ENMs for 25 population estimates for Dipteryx alata, a widely distributed, endemic tree species of the Cerrado region of central Brazil. We propose a new approach for generating a statistical distribution of correlations under randomly generated ENM. The confidence intervals from these distributions indicate how model selection with different properties affects the ability to detect a correlation of interest (e.g. the correlation between He and suitability). Additionally, our approach allows us to explore which particular ensemble of ENMs produces the better result for finding an association between environmental suitability and He. Caution is necessary when choosing a method or a climatic data set for modelling geographical distributions, but the new approach proposed here provides a conservative way to evaluate the ability of ensembles to detect patterns of interest.     

Modelling of continuous microbial cultivation taking into account the memory effects

The problem of chemostat dynamics modelling for the purpose of control is considered. The "memory" of the culture is explicitly taken into account. Two possibilities for improving the quality of the proposed modelling approaches are discussed. A general model that accounts for the culture `memory' by means of different `memory' functions in the expressions of the specific growth rate and of the specific consumption rate and a polynomial function of the substrate concentration for the yield factor is proposed. The case where the maintenance energy is taken into account is also discussed. Two modifications of the general model (w-type and S-type) are presented. A zero-order `memory' function and a i-function with delay are applied in order to describe the `memory' effects. Continuous growth of the strain Saccharomyces cerevisiae on a glucose limited medium is considered as a case study. Detailed investigations of the variety of models, derived from the general model by applying different `memory' functions and different assumptions are carried out. The results are compared with those previously reported for the same process. It is shown that a significant improvement in predicting the substrate dynamics (not accompanied by any decrease in the quality of the model with respect to the biomass concentration) could be achieved, involving a first- or second-order polynomial function for the yield factor. It is also shown that the quality of the model mainly depends on the way that `memory' function is incorporated. The detailed investigations give priority to the w-type models. In this case past values of both biomass and substrate variables are considered. The time delay models with pure (constant) delay and those which account for the culture `memory' by zero-order `memory' function (adaptability parameter) are compared with respect to their utilization for the purpose of model-based control.     

A Reed-Frost model taking into account uncertainties in the diagnostic of the infection

In this paper, we model the epidemic course of a pathogen infection within a semi-closed group which generates clinical signals which do not necessarily permit its ready and certain identification. Typical examples of such a pathogen are influenza-type viruses. We allow for time-varying infectivity levels among individuals, and model the probability of infection per contact as a function of the clinical signals. In order to accomplish this, we introduce a modified chain-binomial Reed-Frost model. We obtain an expression for the basic reproduction ratio and determine conditions which guarantee that the epidemic does not survive in the long-term. These conditions being functions of the signal’s distribution, they can be used to design and evaluate interventions, such as treatment protocols.     

Two mathematical models for predicting dispersion of particles in the human lung

The dispersion of particles in the human lung is modeled as a series of virtual mixing tanks. Using the experimental results of Scherer et al. (1975, J. Appl. Physiol., 38(4), pp. 719-723) for a five-generation glass lung model, it is shown that each generation of the glass lung behaves like an independent virtual mixing tank. The corresponding resident time distribution is shown to have a variance approximately equal to the square of the average time a particle spends in the generation. By assuming that each generation of the human lung behaves as an independent virtual mixing tank, the realistic lung data provided by Weibel (1963, Morphometry of the Human Lung, Spinger-Verlag, New York) are used to validate this assumption in two ways. First, the half-width of the exhaled particle concentration profile is obtained. Second, a system of differential equations, with the concentration of particles in each mixing tank as its solution, is derived and solved numerically. This gives the exhaled concentration profile. Both techniques yield similar results to each other, and both give excellent agreement with the experimental data. The virtual mixing tank approach allows the complex mixing that occurs in the branching pathways of the lung to be more simply modeled. The model, thereby derived, is simple to change and could lead to enhancements in the understanding of the underlying processes contributing to the ventilation of the lung in health and disease.     

Spatially explicit Bayesian clustering models in population genetics

This article reviews recent developments in Bayesian algorithms that explicitly include geographical information in the inference of population structure. Current models substantially differ in their prior distributions and background assumptions, falling into two broad categories: models with or without admixture. To aid users of this new generation of spatially explicit programs, we clarify the assumptions underlying the models, and we test these models in situations where their assumptions are not met. We show that models without admixture are not robust to the inclusion of admixed individuals in the sample, thus providing an incorrect assessment of population genetic structure in many cases. In contrast, admixture models are robust to an absence of admixture in the sample. We also give statistical and conceptual reasons why data should be explored using spatially explicit models that include admixture.     

Some one-dimensional migration models in population genetics theory

This paper gives a method for calculating the covariance of gene frequencies at any two points of a habitat which is a finite 1-dimensional interval. Selectively neutral genes are considered migrating according to a continuous parameter analogue of the stepping stone model. The method is to solve a partial differential equation for the covariance, together with boundary conditions of zero normal derivative (or reflecting barrier) type.     

Theoretical framework of population genetics with somatic mutations taken into account: application to copy number variations in humans

Traditionally, population genetics focuses on the dynamics of frequencies of alleles acquired by mutations on germ-lines, because only such mutations are heritable. Typical genotyping experiments, however, use DNA from some somatic tissues such as blood, which harbors somatic mutations at the current generation in addition to germ-line mutations accumulated since the most recent common ancestor of the sample. This common practice may sometimes cause erroneous interpretations of polymorphism data, unless we properly understand the role of somatic mutations in population genetics. We here introduce a very basic theoretical framework of population genetics with somatic mutations taken into account. It is easy to imagine that somatic mutations at the current generation simply add individual-specific variations, as errors in mutation detection do. Our theory quantifies this increment under various conditions. We find that the major contribution of somatic mutations plus errors is to very rare variants, particularly to singletons. The relative contribution is markedly large when mutations are deleterious. Because negative selection also increases rare variants, it is important to distinguish the roles of these mutually confounding factors when we interpret the data, even after correcting for demography. We apply this theory to human copy number variations (CNVs), for which the composite effect of somatic mutations and errors may not be negligible. Using genome-wide CNV data, we demonstrate how the joint action of the two factors, selection and somatic mutations plus errors, shapes the observed pattern of polymorphism.     

The role of mathematical models in helminth population biology

This brief review aims to illustrate how theory can aid in our understanding of the factors that determine the regulation and stability of parasite abundance, and influence the impact of control measures. The current generation of models are obviously crude, and ignore much biological detail, but they are often able to capture qualitative trends observed in real communities. As such, their analysis and investigation can provide important conceptional insights or, in some circumstances, they can be of value in a predictive role (e.g. the impact of chemotherapy in human communities).This field of research, however, is still in its infancy and much remains to be done to improve biological realism in model formulation and to extent the methods of analysis and interpretation. In the latter context, for example, the current analytical methods for the study of the dynamical properties of non-linear systems of differential and partial differential equations are inadequate for many areas of biological application. Future advances in applied mathematics will, therefore, be of great importance. As far as biological realism is concerned, three areas require urgent attention. The first concerns the treatment of heterogeneity in worm loads within host communities. The generative factors of parasite aggregation are many and varied and little is understood at present of how these processes influence a parasite's population response to perturbation induced, for example, by control measures. Stochastic models are required to examine this problem but current work in this area is very limited.The second area concerns immunity to parasitic infection. Few models take account of the substantive body of experimental work which attests to the significance of host responses (both specific and non-specific) to parasite invasion as determinants of parasite abundance within both an individual host and in the community at large. A start has been made in the investigation of models which mimic acquired immunity and immunological “memory” but much refinement and elaboration is needed (Anderson &; May, 1985a). In particular, the next generation of models should address the details of antibody-antigen and cell-antigen interactions in individual hosts as well as the broader questions concerning herd immunity. Heterogeneity in immunological responsiveness as a consequence of host nutritional status or genetic background must also be condsidered.The final topic is that of population genetics. Geneticists invariably consider changes in gene frequencies without reference to changes in parasite or host abundance, ecologists and epidemiologists have tended to study changes in abundance without reference to changes in genetic structure while immunologists have focused on the mechanisms of resistance to parasitic infection without reference to population or genetic changes. It is becoming increasingly apparent that host genetic background and genetic heterogeneity within parasite populations (e.g. the malarial parasites of man) are important determinants of observed population events (Medley &; Anderson, 1985). Future research must attempt to meld the areas of genetics, population dynamics and immunology. Such an integration presents a fascinating challenge.     

Interactive use of computer models in teaching population genetics

A technique using a computer with a graphical display unit to teach students the effects of genetic drift, selection and migration is described. Both diallelic and triallelic loci are discussed. The Fokker-Planck equation is used as the mathematical model of the genetic system, and its validity as an approximation in this context is demonstrated by an investigation into selection at the ABO locus. An appendix contains a derivation from the Fokker-Planck equation of the formula used in the paper for the gene frequency distribution at a multiallelic locus in equilibrium under selection and migration.     

Projection of cancer risks from the Japanese atomic bomb survivors to the England and Wales population taking into account uncertainty in risk parameters

Generalized relative risk models, with adjustments to the relative risk for time after exposure and age at exposure and incorporating a linear-quadratic dose response, were fitted to the latest (Life Span Study Report 12) Japanese atomic bomb survivor cancer mortality data using Bayesian Markov Chain Monte Carlo methods, taking account of random errors in the DS86 dose estimates. The resulting uncertainty distributions in the relative risk model parameters were used to derive uncertainties in population cancer risks for a current UK population. Following an assumed administered dose of 1 Sv, leukaemia mortality risks were estimated to be 1.93×10^–2 Sv^–1 (95% CI 1.14, 3.38), or 0.44 years of life lost Sv^–1 (95% CI 0.22, 0.94). Following an assumed administered dose of 1 Sv, solid cancer mortality risks were calculated to be 10.36×10^–2 Sv^–1 (95% CI 8.41, 12.42), or 1.38 years of life lost Sv^–1 (95% CI 1.11, 1.68). In general, solid cancer risks were very similar to those predicted by classical likelihood-based methods; however, leukaemia risks were somewhat higher, by 10–35%, than those predicted by classical likelihood-based methods. This is so in both cases, irrespective of whether or not adjustments are made in these likelihood-based fits for the effects of measurement errors, and the discrepancy for leukaemia tends to be greater at higher doses. Overall, cancer risks predicted by Bayesian Markov Chain Monte Carlo methods are similar to those derived by classical likelihood-based methods and which form the basis of established estimates of radiation-induced cancer risk. Received: 28 September 1999 / Accepted: 21 August 2000     

Development of the cephalosporin C fermentation taking into account the instability of cephalosporin C

Summary Cephalosporin C undergoes chemical hydrolysis during fermentation at 25°C at a rate of 0.48% per hour to give increasing amounts of 2-(D-4-amino-4-carboxybutyl)thiazole-4-carboxylic acid (compound X). Although the cephalosporin C titer in a fermentation levels off at 160–170h, the total of C+X rises almost linearly up to 200h. Cells, therefore, biosynthesize cephalosporin C at an almost constant rate up to 200h. This instability greatly influences attempts to improve the cephalosporin strain by genetic manipulation, or the process by alterations in technology.