Dependence of biological treatment rate on species composition in activated sludge or biofilm. II: From models to theory

Traditional biological treatment models are "deduced" from formal chemical kinetics or dynamics of pure microorganism cultures growth. The best formal models give reasonable approximations of the biological treatment model with an ecosystem adaptation (ESA model). The model presented here explains some features of the biological treatment mechanism that cannot be described by formal models.     

Characterization of commercial and biological growth curves in the Segureña sheep breed

Non-linear models were analysed to describe both the biological and commercial growth curves of the Segureña sheep, one of the most important Spanish breeds. We evaluated Brody, von Bertalanffy, Verhulst, logistic and Gompertz models, using historical data from the National Association of Segureña Sheep Breeders (ANCOS). These records were collected between 2000 and 2013, from a total of 129 610 weight observations ranging from birth to adulthood. The aim of this research was to establish the mathematical behaviour of body development throughout this breed’s commercial life (birth to slaughter) and biological life (birth to adulthood); comparison between both slopes gives important information regarding the best time for slaughter, informs dietary advice according to animals’ needs, permits economical predictions of productions and, by using the curve parameters as selection criteria, enables improvements in growth characteristics of the breed. Models were fitted according to the non-linear regression procedure of statistical package SPSS version19. Model parameters were estimated using the Levenberg–Marquardt algorithm. Candidate models were compared using the determinative coefficient, mean square error, number of iterations, Akaike information coefficient and biological coherence of the estimated parameters. The von Bertalanffy and logistic models were found to be best suited to the biological and commercial growth curves, respectively, for both sexes. The Brody equation was found to be unsuitable for studying the commercial growth curve. Differences between the parameters in both sexes indicate a strong impact of sexual dimorphism on growth. This can emphasize the value of the highest growth rate for females, indicating that they reach maturity earlier.     

Probabilistic Gompertz model of irreversible growth

Characterizing organism growth within populations requires the application of well-studied individual size-at-age models, such as the deterministic Gompertz model, to populations of individuals whose characteristics, corresponding to model parameters, may be highly variable. A natural approach is to assign probability distributions to one or more model parameters. In some contexts, size-at-age data may be absent due to difficulties in ageing individuals, but size-increment data may instead be available (e.g., from tag-recapture experiments). A preliminary transformation to a size-increment model is then required. Gompertz models developed along the above lines have recently been applied to strongly heterogeneous abalone tag-recapture data. Although useful in modelling the early growth stages, these models yield size-increment distributions that allow negative growth, which is inappropriate in the case of mollusc shells and other accumulated biological structures (e.g., vertebrae) where growth is irreversible. Here we develop probabilistic Gompertz models where this difficulty is resolved by conditioning parameter distributions on size, allowing application to irreversible growth data. In the case of abalone growth, introduction of a growth-limiting biological length scale is then shown to yield realistic length-increment distributions.     

Hyperbolastic growth models: theory and application

Background  

Mathematical models describing growth kinetics are very important for predicting many biological phenomena such as tumor volume, speed of disease progression, and determination of an optimal radiation and/or chemotherapy schedule. Growth models such as logistic, Gompertz, Richards, and Weibull have been extensively studied and applied to a wide range of medical and biological studies. We introduce a class of three and four parameter models called "hyperbolastic models" for accurately predicting and analyzing self-limited growth behavior that occurs e.g. in tumors. To illustrate the application and utility of these models and to gain a more complete understanding of them, we apply them to two sets of data considered in previously published literature.     

Simulation of biological growth

Researchers concerned with the growth of biological tissue often use models that predict the growth as a function of a mechanical stimulus such as stress, strain or elastic energy. However, a general theory for bulk growth should consider that the mechanical stimulus may only be one of many factors contributing to growth. Another important factor could be time, as living tissues can be assumed to have a pre-programmed directional biological growth that is independent of mechanical stimuli. This paper has two objectives: the first is to introduce the concept of directional biological growth within a well developed growth theory, the second is to present the computational methods by which three-dimensional growth that encompasses time and stress effects can be simulated using commercially available finite element analysis software.     

Analysis of logistic growth models

A variety of growth curves have been developed to model both unpredated, intraspecific population dynamics and more general biological growth. Most predictive models are shown to be based on variations of the classical Verhulst logistic growth equation. We review and compare several such models and analyse properties of interest for these. We also identify and detail several associated limitations and restrictions.A generalized form of the logistic growth curve is introduced which incorporates these models as special cases. Several properties of the generalized growth are also presented. We furthermore prove that the new growth form incorporates additional growth models which are markedly different from the logistic growth and its variants, at least in their mathematical representation. Finally, we give a brief outline of how the new curve could be used for curve-fitting.     

Predicting local biological characteristics in streams: a comparison of landscape classifications

1. Landscape classifications group tracts of land based on similar physico‐chemical attributes that may affect the biological characteristics of streams at local scales. We tested the ability of five landscape models to account for variation in algal and macroinvertebrate biomass, brook trout (Salvelinus fontinalis) growth and macroinvertebrate community composition from 132 riffles in 15 catchments in the Big Horn Mountains, Wyoming. 2. A model created by the U.S. Forest Service (FS) combined catchment and ecoregion approaches to classify the landscape. Our model used digital elevations to create a landscape classification for streams (DEM). The last three models were based on: (1) standard ecoregions (Ecoregion), (2) the type of underlying bedrock (Geology) and (3) the geographical distance between sites (Site Proximity model). 3. Overall, the Ecoregion and Geology models performed better than the two catchment models (FS and DEM) in predicting local biological characteristics. The Geology model was best at predicting differences in algal and macroinvertebrate biomass, the Site Proximity and Ecoregion models were best at predicting patterns of similarity in macroinvertebrate community composition, and the Site Proximity, Ecoregion, Geology, and FS models, in order from best to worst, accounted for significant variation in brook trout growth. The Site Proximity model performed well because of the effects of spatial autocorrelation. The DEM was consistently one of the worst models at predicting local biological characteristics because it failed to include important attributes (e.g. dominant geology). Calcareous geology was positively associated with greater macroinvertebrate biomass and faster brook trout growth, but it was inversely related to algal biomass. 4. None of the models accounted for a large amount of variation in local biological characteristics. Single‐scale, landscape classifications may never accurately predict variation in local biological characteristics because: (1) landscapes show a high degree of spatial heterogeneity, (2) local effects are stronger than landscape attributes and (3) there are too many intervening levels between landscape and local scales in the nested hierarchy of streams. However, landscape classifications did account for significant variation in biological characteristics. Thus, they would be a valuable management tool as part of a multi‐scale, hierarchical technique for classifying stream ecosystems.     

Module-based multiscale simulation of angiogenesis in skeletal muscle

Background  

Mathematical modeling of angiogenesis has been gaining momentum as a means to shed new light on the biological complexity underlying blood vessel growth. A variety of computational models have been developed, each focusing on different aspects of the angiogenesis process and occurring at different biological scales, ranging from the molecular to the tissue levels. Integration of models at different scales is a challenging and currently unsolved problem.     

Dynamics of termite populations-some theoretical considerations

Summary Various mathematical models are examined to fit the growth of termite populations. Termite populations show marked periodic fluctuations and these have their own biological significance. In the long run, notwith-standing the periodic fluctuations, the population continues to grow, if not infinitely.The intrinsic rate of population growth is dependent primarily on the fecundity of the queen. The population levels of different castes and population fluctuations are brought about by the selective development of the eggs into the adults of different castes and alates. Most of the models fail to account for the physiological and biological factors involved in termite population growth. A tentative model is suggested to express the population growth against time.Several hypotheses concerning the population regulation mechanisms of insects are examined and evaluated in relation to the termite populations. It is suggested that population growth of termites is probably regulated by homeostatic mechanisms.     

The exact probabilities of branching patterns under terminal and segmental growth hypotheses

Information about the way of branching of dendritic arborizations may be obtained by comparing the frequency distributions of observed branching patterns with theoretical distributions based on well-defined growth models. Two models usually get much attention in geomorphological and (neuro)biological studies, viz. terminal growth and segmental growth. Formulae to construct the exact probability distributions for both growth models are presented. It is shown that ranking and lumping of the individual branching patterns enable the analysis of very large arborizations with relatively few data. The application of the Kolmogorov goodness-of-fit test for discrete distributions to the analysis is discussed.     

水稻自然生长趋势的数学模型和动态分析

本研究以随机过程理论为基础,应用数理方法,在分期播种的基础上,提出了一个受遗传特性控制的水稻各生物学参量自然生长趋势的数学模型。并进行了重复验证,得到了不同生育时段各生物学参量的生长速度（一阶导数）和生长速度的变化率（二阶导数）,揭示了水稻生长的基本规律。并结合实测资料,通过相关分析,得到影响各生物学参量的关键气象因子,将水稻生长的研究建立在定量的基础上,有利于更加合理地采用各项高产栽培和农田管理措施。     

Predictions of response to temperature are contingent on model choice and data quality

The equations used to account for the temperature dependence of biological processes, including growth and metabolic rates, are the foundations of our predictions of how global biogeochemistry and biogeography change in response to global climate change. We review and test the use of 12 equations used to model the temperature dependence of biological processes across the full range of their temperature response, including supra‐ and suboptimal temperatures. We focus on fitting these equations to thermal response curves for phytoplankton growth but also tested the equations on a variety of traits across a wide diversity of organisms. We found that many of the surveyed equations have comparable abilities to fit data and equally high requirements for data quality (number of test temperatures and range of response captured) but lead to different estimates of cardinal temperatures and of the biological rates at these temperatures. When these rate estimates are used for biogeographic predictions, differences between the estimates of even the best‐fitting models can exceed the global biological change predicted for a decade of global warming. As a result, studies of the biological response to global changes in temperature must make careful consideration of model selection and of the quality of the data used for parametrizing these models.     

The growth and the fluid dynamics of protein crystals and soft organic tissues: models and simulations,similarities and differences

The fluid-dynamic environment within typical growth reactors as well as the interaction of such flow with the intrinsic kinetics of the growth process are investigated in the frame of the new fields of protein crystal and tissue engineering. The paper uses available data to introduce a set of novel growth models. The surface conditions are coupled to the exchange mass flux at the specimen/culture-medium interface and lead to the introduction of a group of differential equations for the nutrient concentration around the sample and for the evolution of the construct mass displacement. These models take into account the sensitivity of the construct/liquid interface to the level of supersaturation in the case of macromolecular crystal growth and to the "direct" effect of the fluid-dynamic shear stress in the case of biological tissue growth. They then are used to show how the proposed surface kinetic laws can predict (through sophisticated numerical simulations) many of the known characteristics of protein crystals and biological tissues produced using well-known and widely used reactors. This procedure provides validation of the models and associated numerical method and at the same time gives insights into the mechanisms of the phenomena. The onset of morphological instabilities is discussed and investigated in detail. The interplay between the increasing size of the sample and the structure of the convective field established inside the reactor is analysed. It is shown that this interaction is essential in determining the time evolution of the specimen shape. Analogies about growing macromolecular crystals and growing biological tissues are pointed out in terms of behaviours and cause-and-effect relationships. These aspects lead to a common source (in terms of original mathematical models, ideas and results) made available for the scientific community under the optimistic idea that the contacts established between the "two fields of engineering" will develop into an ongoing, mutually beneficial dialogue.     

Organic tissues in rotating bioreactors: fluid-mechanical aspects, dynamic growth models, and morphological evolution

This analysis deals with advances in tissue-engineering models and computational methods as well as with novel results on the relative importance of "controlling forces" in the growth of organic constructs. Specifically, attention is focused on the rotary culture system, because this technique has proven to be the most practical solution for providing a suitable culture environment supporting three-dimensional tissue assemblies. From a numerical point of view, the growing biological specimen gives rise to a moving boundary problem. A "volume-of-fraction" method is specifically and carefully developed according to the complex properties and mechanisms of organic tissue growth and, in particular, taking into account the sensitivity of the construct/liquid interface to the effect of the fluid-dynamic shear stress (it induces changes in tissue metabolism and function that elicit a physiological response from the biological cells). The present study uses available data to introduce a set of growth models. The surface conditions are coupled to the transfer of mass and momentum at the specimen/culture-medium interface and lead to the introduction of a group of differential equations for the nutrient concentration around the sample and for the evolution of tissue mass displacement. The models are then used to show how the proposed surface kinetic laws can predict (through sophisticated numerical simulations) many of the known characteristics of biological tissues grown using rotating-wall perfused vessel bioreactors. This procedure provides a validation of the models and associated numerical method and also gives insight into the mechanisms of the phenomena. The interplay between the increasing size of the tissue and the structure of the convective field is investigated. It is shown that this interaction is essential in determining the time evolution of the tissue shape. The size of the growing specimen plays a critical role with regard to the intensity of convection and the related shear stresses. Convective effects, in turn, are found to impact growth rates, tissue size, and morphology, as well as the mechanisms driving growth. The method exhibits novel capabilities to predict and elucidate experimental observations and to identify cause-and-effect relationships.     

State-structure models of microbial growth

A framework for modelling microbial growth is given in which the biological properties of the regulation of cell growth and proliferation are taken as a basis. Some stages on the way of formulating state-structure models are discussed. The resulting delay-differential equations are both comprehensible and suitable for a mathematical treatment.     

Mathematical Models of the Kinetics of the Cultivation of Microorganisms

Various equations of mathematical models for the kinetics of the development of various biological processes were obtained on the basis of the generalized differential equation of biomass growth. Aerobic periodic cultivation of the yeast Saccharomyces cerevisiae was carried out to provide a comparative evaluation of advantages and disadvantages of four types of mathematical models. It is shown that the exponential model is a particular solution to the generalized differential equation. The developed mathematical model can be used to predict the course of biological processes in time and can serve as a tool for a computational experiment in order to clarify the dependence of the rate of a biological process on changes in certain parameters that affect the development of cells.     

Revisiting Verhulst and Monod models: analysis of batch and fed-batch cultures

The paper re-evaluates Verhulst and Monod models. It has been claimed that standard logistic equation cannot describe the decline phase of mammalian cells in batch and fed-batch cultures and in some cases it fails to fit somatic growth data. In the present work Verhulst, population-based mechanistic growth model was revisited to describe successfully viable cell density (VCD) in exponential and decline phases of batch and fed-batch cultures of three different CHO cell lines. Verhulst model constants, K, carrying capacity (VCD/ml or μg/ml) and r, intrinsic growth factor (h⁻¹) have physical meaning and they are of biological significance. These two parameters together define the course of growth and productivity and therefore, they are valuable in optimisation of culture media, developing feeding strategies and selection of cell lines for productivity. The Verhulst growth model approach was extended to develop productivity models for batch and fed-batch cultures. All Verhulst models were validated against blind data (R² > 0.95). Critical examination of theoretical approaches concluded that Monod parameters have no physical meaning. Monod-hybrid (pseudo-mechanistic) batch models were validated against specific growth rates of respective bolus and continuous fed-batch cultures (R² ≈ 0.90). The reduced form of Monod-hybrid model C_L/(K_L + C_L) describes specific growth rate during metabolic shift (R² ≈ 0.95). Verhulst substrate-based growth models compared favourably with Monod-hybrid models. Thus, experimental evidence implies that the constants in the Monod-hybrid model may not have physical meaning but they behave similarly to the biological constants in Michaelis–Menten enzyme kinetics, the basis of the Monod growth model.     

Modeling arbuscular mycorrhizal infection: is % infection an appropriate variable?

Several available models of arbuscular mycorrhizal infection are based on fitting % infection to a logistic curve and then relating the various parameters to biological functions. I suggest here that this direction is misleading. Percent infection is a value derived from the growth of two interdependent but distinct organisms, each of which is seeking to maximize its own growth and survival. I suggest that two-organism models, such as those derived from Lotka-Volterra equations, are more useful for understanding the biology and functioning of mycorrhizae. Accepted: 22 October 2000     

Scaling from mycelial growth to infection dynamics: a reaction diffusion approach

Numerous models have been proposed for the dynamics of fungal growth, and also for the dynamics of infection. Few models, however, have combined the mechanistic interpretation of mycelial growth with epidemiological models for the transmission of infection. Many of the mechanistic models seek to include considerable biological detail, which necessarily leads to a proliferation of state variables and parameters. Including such models within an epidemiological framework makes interpretation of underpinning processes difficult. A simple reaction diffusion model for the growth and spread of fungal mycelium is introduced and analysed, scaling from the small-scale parameters for mycelial dynamics to the large-scale properties of the colony. By coupling the output to a parsimonious epidemiological model for the dynamics of primary infection, we analyse the sensitivity of the probability of successful infection of a host to the colony dynamics associated with local bulking-up, extension, growth and nutrient consumption by the mycelium. In particular we identify optimal trade-offs in bulking-up versus dispersal in controlling infection dynamics.