Growth in solid heterogeneous human colon adenocarcinomas: comparison of simple logistical models

Three models of simple logistical growth were used to describe volumetric growth in heterogeneous tumours. Two clonal subpopulations (designated as clone A and clone D) originally obtained from a human colon adenocarcinoma were used to produce solid xenograft tumours in nude mice. Volumetric growth of tumours produced from pure cells alone was compared to that produced from 50% A:50% D, 88% A:12% D, and 9% A:91% D admixtures. Gompertzian analysis of the in vivo growth data indicated significant differences in both the initial growth rates and final asymptotic limiting volumes of the pure versus the admixed tumours. Verhulstian and modified Verhulstian models were also used to derive regression curves from the same data. The fit of the curves was compared with each other using standard (Akaike, 1974; Schwartz, 1978) information criteria. In four of the five tumour populations the Gompertz equation fitted best. Only in the 88% A:12% D tumours did the modified Verhulst model fit best. The deviations from the regression curves, the residuals, for all three models were systematically distributed. These systematic errors are likely to be the result of using simplified logistical models to describe the growth kinetics of interacting populations in heterogeneous tumours.     

Gompertzian re-evaluation of the growth patterns of transplantable mammary tumours in sialoadenectomized mice

Abstract
The method of the recursion formula of the Gompertz function (Bassukas & Maurer-Schultze 1988) has been applied to analyse tumour growth data taken from the literature; namely the growth perturbation of transplantable mammary tumours in sialoadenectomized mice with or without subsequent epidermal growth factor substitution (results on two mouse strains, C3H or SHN, have been reported; Inui, Tsubura & Morii 1989). The recursion formula of the Gompertz function fits growth curves to all seven sets of data well ( P > 0.05 for lack of fit test). The growth pattern of the tumours in the unperturbed hosts is Gompertzian and does not change if tumours are transplanted in sialoadenectomized mice, although the starting specific growth rate decreases in C3H mice. However, if sialoadenectomy is carried out after tumour inoculation, a complex alteration of the tumour growth evolves: tumour growth does not simply decelerate but it also shifts from the conventional Gompertzian to an exponential or even 'hyperexpo-nential' growth pattern, i.e. with an accelerating specific growth rate. Some theoretical mechanisms of this alteration, as well as the differences between the present Gompertzian analysis and a previously published Verhulstian analysis of part of the same data (Leith, Harrigan & Michelson 1991), are discussed. It is concluded that the quantitative analysis of tumour growth patterns by the method of the difference equation of the Gompertz function presently applied may substantially contribute to the improvement of the interpretation of perturbations of tumour growth-irrespective of their genesis. In contrast to the application of some a priori fixed growth function, e.g. the Verhulstian one, the present method can quantitatively interprete different growth patterns and their classification on the basis of linear regression analysis.     

Modelling Biolog profiles' evolution for yeast growth monitoring in alcoholic fermentation

Aims: A research was undertaken to explore the possibility to express with suitable mathematical models Biolog metabolic curves obtained for oenological yeasts and to use such models for monitoring yeast growth in alcoholic fermentation. Methods and Results: Experimental curves of metabolic activity in Biolog YT microplates, obtained in a previous work for various oenological yeast strains in pure cultures and mixed populations, at various cell concentrations, have been modelled with Gompertz’s, Gompertz’s modified and Lindstrom’s mathematical equations. Lindstrom’s model proved to be the most suitable to fit the curves of the oenological yeasts under study, providing the highest correlation coefficients between experimental and calculated data. The model made it possible to recognize, in mixed yeast populations, the presence of active dry yeasts used for guided fermentations. Model’s constant parameters were used for a numerical characterization of yeast curves. Conclusions: The application of the model to the experimental data resulted to be suitable for an early prediction of the successive evolution of yeast growth. Significance and Impact of the Study: The results obtained indicate the possibility to develop protocols for monitoring yeast presence during alcoholic fermentation, with an early assessment of the correct evolution of their growth, especially when active dry yeasts are employed.     

Tumor micro-ecology and competitive interactions

Three nested models describing the growth of individual subpopulations in a heterogeneous environment are described. The models represent the dynamics of two populations which compete, to varying degrees, for common resources. The first model describes growth in a totally non-competitive micro-environment, the second model describes an ecology in which competition is proportional to competitor population size, and the third model ecology extends the model described by Jansson & Revesz (1974), which allows one population to emerge from the other. The critical points for each model are defined using the isoclines derived from the Ordinary Differential Equations (ODE's) describing competitive growth. The critical points for each model are characterized by the signs of the eigenvalues of the variational matrix at each point. The theoretical results of the analysis show that a competitive model ecology with Verhulstian logistics allows four critical points: the origin which is a repeller, two competitive exclusion points, and an equilibrium state (Waltman, 1983). The extended model ecology of Jansson & Revesz (1974), allows three critical points: the origin which is a repeller, competitive exclusion of the first population, and an equilibrium point. Data from a human adenocarcinoma of the colon and murine mammary tumors are used as qualitative measures of the dynamics of the three micro-ecologies. Issues such as stochastic extension to model small populations either for clonal extinction or heterogeneous emergence are discussed.     

Flow cytometric and karyotypic analyses of androgen-responsive and androgen-independent Shionogi mouse mammary tumours

The androgen-responsive (AR) Shionogi mouse mammary carcinoma is a heterogeneous tumour composed of AR and androgen-independent (AI) cells. We characterized the cells of the AR tumour and those of its AI derivative by flow cytometric analysis of DNA content and karyotypic analysis of metaphase spreads. Both tumours had diploid and near tetraploid populations of cells. However, the AR and AI malignant cells of these tumours both appeared to be polyploid. A decrease in the polyploid population of the AR tumour accompanied tumour regression following castration, but this population was restored when tumour growth resumed. Although karyotypic analysis of metaphase spreads showed wide variations in chromosome numbers among the polyploid population, the range, 55-88 chromosomes, was found in both AR and AI tumours. In addition, the same chromosome anomalies, including a marker chromosome, were identified in both tumours. Since the AR and AI malignant cells could not be distinguished on the basis of their DNA content or karyotype, the cell types may not represent genetically distinct populations of cells. The AR cells may undergo alterations in gene expression in adapting to their androgen-free environment.     

Mathematical models of bacterial growth,inhibition and death under combined stress conditions

Summary In this report we review the history of growth theories. We show how classical growth models may be derived as special cases of a generic growth rate equation. We show how growth models may be modified to represent survival data. We use linear combinations of growth and survival models to represent complex growth/survival curves and give practical examples utilizing nonlinear regression analysis. We show that traditional methods of estimating D values are inappropriate for complex, multiphasic growth/survival data. We show how such data may be modeled mathematically and illustrate methods for estimating true D values from such data.     

Discovery of Power-Law Growth in the Self-Renewal of Heterogeneous Glioma Stem Cell Populations

Background

Accumulating evidence indicates that cancer stem cells (CSCs) drive tumorigenesis. This suggests that CSCs should make ideal therapeutic targets. However, because CSC populations in tumors appear heterogeneous, it remains unclear how CSCs might be effectively targeted. To investigate the mechanisms by which CSC populations maintain heterogeneity during self-renewal, we established a glioma sphere (GS) forming model, to generate a population in which glioma stem cells (GSCs) become enriched. We hypothesized, based on the clonal evolution concept, that with each passage in culture, heterogeneous clonal sublines of GSs are generated that progressively show increased proliferative ability.

Methodology/Principal Findings

To test this hypothesis, we determined whether, with each passage, glioma neurosphere culture generated from four different glioma cell lines become progressively proliferative (i.e., enriched in large spheres). Rather than monitoring self-renewal, we measured heterogeneity based on neurosphere clone sizes (#cells/clone). Log-log plots of distributions of clone sizes yielded a good fit (r>0.90) to a straight line (log(% total clones) = k*log(#cells/clone)) indicating that the system follows a power-law (y = x^k) with a specific degree exponent (k = −1.42). Repeated passaging of the total GS population showed that the same power-law was maintained over six passages (CV = −1.01 to −1.17). Surprisingly, passage of either isolated small or large subclones generated fully heterogeneous populations that retained the original power-law-dependent heterogeneity. The anti-GSC agent Temozolomide, which is well known as a standard therapy for glioblastoma multiforme (GBM), suppressed the self-renewal of clones, but it never disrupted the power-law behavior of a GS population.

Conclusions/Significance

Although the data above did not support the stated hypothesis, they did strongly suggest a novel mechanism that underlies CSC heterogeneity. They indicate that power-law growth governs the self-renewal of heterogeneous glioma stem cell populations. That the data always fit a power-law suggests that: (i) clone sizes follow continuous, non-random, and scale-free hierarchy; (ii) precise biologic rules that reflect self-organizing emergent behaviors govern the generation of neurospheres. That the power-law behavior and the original GS heterogeneity are maintained over multiple passages indicates that these rules are invariant. These self-organizing mechanisms very likely underlie tumor heterogeneity during tumor growth. Discovery of this power-law behavior provides a mechanism that could be targeted in the development of new, more effective, anti-cancer agents.     

Effect of carbon:nitrogen ratio on kinetics of phenol biodegradation byAcinetobacter johnsonii in saturated sand

In polluted soil or ground water, inorganic nutrients such as nitrogen may be limiting, so that Monod kinetics for carbon limitation may not describe microbial growth and contaminant biodegradation rates. To test this hypothesis we measured¹⁴CO₂ evolved by a pure culture ofAcinetobacter johnsonii degrading 120 µg¹⁴C-phenol per ml in saturated sand with molar carbon:nitrogen (CN) ratios ranging from 1.5 to 560. We fit kinetics models to the data using non-linear least squares regression. Phenol disappearance and population growth were also measured at CN1.5 and CN560.After a 5- to 10-hour lag period, most of the¹⁴CO₂ evolution curves at all CN ratios displayed a sigmoidal shape, suggesting that the microbial populations grew. As CN ratio increased, the initial rate of¹⁴CO₂ evolution decreased. Cell growth and phenol consumption occurred at both CN1.5 and CN560, and showed the same trends as the¹⁴CO₂ data. A kinetics model assuming population growth limited by a single substrate best fit the¹⁴CO₂ evolution data for CN1.5. At intermediate to high CN ratios, the data were best fit by a model originally formulated to describe no-growth metabolism of one substrate coupled with microbial growth on a second substrate. We suggest that this dual-substrate model describes linear growth on phenol while nitrogen is available and first-order metabolism of phenol without growth after nitrogen is depleted.     

Modeling physical growth using mixed effects models

This article demonstrates the use of mixed effects models for characterizing individual and sample average growth curves based on serial anthropometric data. These models are advancement over conventional general linear regression because they effectively handle the hierarchical nature of serial growth data. Using body weight data on 70 infants in the Born in Bradford study, we demonstrate how a mixed effects model provides a better fit than a conventional regression model. Further, we demonstrate how mixed effects models can be used to explore the influence of environmental factors on the sample average growth curve. Analyzing data from 183 infant boys (aged 3–15 months) from rural South India, we show how maternal education shapes infant growth patterns as early as within the first 6 months of life. The presented analyses highlight the utility of mixed effects models for analyzing serial growth data because they allow researchers to simultaneously predict individual curves, estimate sample average curves, and investigate the effects of environmental exposure variables. Am J Phys Anthropol, 2013. © 2012 Wiley Periodicals, Inc.     

Survival of genetically engineered Pseudomonas fluorescens in soil in competition with the parent strain

Abstract The population dynamics of two genetically engineered Pseudomonas fluorescens strains, D5 and C5t, introduced into a loamy sand soil, in competition with a spontaneous antibiotic-resistant mutant of the corresponding wildtype strain was studied. Strain D5 contained an insertion of transposon Tn5 in its genome, whereas strain C5t was obtained by insertion of Tn 5 :: tox , a Tn 5 -derivative containing a Bacillus thuringiensis var. morrisoni δ-endotoxin gene, into the chromosome using a suicide vector system. Southern hybridization analysis demonstrated the absence of vector sequences, and the presence of single copies of either Tn 5 or Tn 5 :: tox in the respective strains. Western blotting and a bio-assay on larvae of Anopheles stephensi suggested the tox gene was functional in clone C5t. Both D5 and C5t were prototrophic and their generation times in minimal medium were slightly below that of the corresponding wild-type strain. Tn 5 and Tn 5 :: tox were stable in both clones during growth in minimal medium for 16 generations. During growth in competition with the wild-type strain, D5 competed well, however C5t was outcompeted from 50 to below 3% of the population in 40 generations. During growth in competition in the sterile loamy sand, both strains were outcompeted by the parent strain; strain C5t was less competitive than D5. In non-sterile loamy sand, the introduced mixed populations showed a slow decline; both C5t and D5 were outcompeted by the parent strain. The decreased fitness of both modified strains, although significant, was considered to be small in ecological terms. Further, the addition of 10% bentonite clay to the loamy sand resulted in a significant enhancement of survival of the mixed populations, and a stabilization of the proportions between the modified strains and the parent. Finally, there was a trend towards a decrease in the proportion modified strain/parent strain in both mixes in the rhizosphere of wheat.     

Fit optimisation of a distal medial tibia plate

An iterative method for the fit optimisation of a pre-contoured fracture fixation plate for a given bone data set is presented. Both plate shape optimisation and plate fit quantification are conducted in a virtual environment utilising computer graphical methods and 3D bone and plate models. Two optimised shapes of the undersurface of an existing distal medial tibia plate were generated based on a dataset of 45 3D bone models reconstructed from computed tomography image data of Japanese tibiae. The existing plate shape achieved an anatomical fit on 13% of tibiae from the dataset. Modified plate 1 achieved an anatomical fit for 42% and modified plate 2 a fit for 67% of the bones. If either modified plate 1 or plate 2 is used, then the anatomical fit can be increased to 82% for the same dataset. Issues pertaining to any further improvement in plate fit/shape are discussed.     

Assessing biodiversity with species accumulation curves; inventories of small reptiles by pit‐trapping in Western Australia

Abstract We examined 11 non‐linear regression models to determine which of them best fitted curvilinear species accumulation curves based on pit‐trapping data for reptiles in a range of heterogeneous and homogenous sites in mesic, semi‐arid and arid regions of Western Australia. A well‐defined plateau in a species accumulation curve is required for any of the models accurately to estimate species richness. Two different measures of effort (pit‐trapping days and number of individuals caught) were used to determine if the measure of effort influenced the choice of the best model(s). We used species accumulation curves to predict species richness, determined the trapping effort required to catch a nominated percentage (e.g. 95%) of the predicted number of species in an area, and examined the relationship between species accumulation curves with diversity and rarity. Species richness, diversity and the proportion of rare species in a community influenced the shape of species accumulation curves. The Beta‐P model provided the best overall fit (highest r²) for heterogeneous and homogeneous sites. For heterogeneous sites, Hill, Rational, Clench, Exponential and Weibull models were the next best. For homogeneous habitats, Hill, Weibull and Chapman–Richards were the next best models. There was very little difference between Beta‐P and Hill models in fitting the data to accumulation curves, although the Hill model generally over‐estimated species richness. Most models worked equally well for both measures of trapping effort. Because the number of individuals caught was influenced by both pit‐trapping effort and the abundance of individuals, both measures of effort must be considered if species accumulation curves are to be used as a planning tool. Trapping effort to catch a nominated percentage of the total predicted species in homogeneous and heterogeneous habitats varied among sites, but even for only 75% of the predicted number of species it was generally much higher than the typical effort currently being used for terrestrial vertebrate fauna surveys in Australia. It was not possible to provide a general indication of the effort required to predict species richness for a site, or to capture a nominated proportion of species at a site, because species accumulation curves are heavily influenced by the characteristics of particular sites.     

十三碳二元酸发酵过程产酸期拟均相动力学模型及其应用

分析了十三碳二元酸发酵过程中产酸期的代谢特点,对产酸期四相体系发酵动力学进行了研究。提出了菌体生长、产物形成及底物消耗的动力学模型,对模型参数进行了回归估值,并对产酸期进行了拟合,结果表明,模型的计算值和实测值较为吻合,平均相对偏差为3．6％。利用所建模型对产酸期进行了多种操作条件下的模拟计算,结果表明,提高进入产酸期的菌体浓度、缩短菌体生长期时间及降低发酵液中产物浓度具有提高产物形成速率的有效途径。     

光响应曲线的指数改进模型与常用模型比较

光响应曲线的参数是研究植物生理状态的重要指标, 常用的光响应曲线模型无法准确地计算出光饱和点和最大净光合速率。该文利用光响应曲线新模型——指数改进模型、直角双曲线模型、直角双曲线修正模型、非直角双曲线模型和指数模型, 拟合高粱(Sorghum bicolor)、苋(Amaranthus tricolor)、大麦(Hordeum vulgare)和半夏(Pinellia ternata)的光响应曲线, 并随机选取部分数据进行检验, 得到了各模型计算出的主要生理参数, 并对这些数据进行了比较分析, 讨论了各模型之间的优缺点和准确性, 描述了C₃、C₄植物光响应的适宜性。结果表明, 基于C₃植物得到的指数改进模型和直角双曲线修正模型能较准确地计算出C₃、C₄植物饱和光强和最大净光合速率, 并在描述光响应曲线时比另外3个模型具有更高的精确性和适宜性。实验结果可为光响应曲线模型在C₃和C₄光合途径植物中的应用提供参考。     

Modeling the growth of Enterococcus faecium in bologna sausage.

A study to set up mathematical models which allow the prediction of Enterococcus faecium growth in bologna sausage (mortadella) was carried out. Growth curves were obtained at different temperatures (5, 6, 12, 15, 25, 32, 35, 37, 42, 46, 50, 52, and 55 degrees C). The Gompertz and logistic models, modified by Zwietering, were found to fit with the representation of experimental curves. The variations of the parameters A (i.e., the asymptotic value reached by the relative population during the stationary growth phase), mu m (i.e., the maximum specific growth rate during the exponential growth phase), and lambda (i.e., the lag time) with temperature were then modeled. The variation of A with temperature can be described by an empirical polynomial model, whereas the variation of mu m and lambda can be described by the Ratkowsky model modified by Zwietering and the Adair model, respectively. Data processing of these models has shown that the minimum growth temperature for E. faecium is 0.1 degrees C, the maximum growth temperature is 53.4 degrees C, and the optimal growth temperature is 42 to 45 degrees C.     

Analyzing population growth curves

Assessing animal population growth curves is an essential feature of field studies in ecology and wildlife management. We used five models to assess population growth rates with a number of sets of population growth rate data. A 'generalized' logistic curve provides a better model than do four other popular models. Use of difference equations for fitting was checked by a comparison of that method and direct fitting of the analytical (integrated) solution for three of the models. Fits to field data indicate that estimates of the asymptote, K, from the 'generalized logistic' and the ordinary logistic agree well enough to support use of estimates of K from the ordinary logistic on data that cannot be satisfactorily fitted with the generalized logistic. Akaike's information criterion is widely used, often with a small sample version AIC_c. Our study of five models indicated a bias in the AIC_c criterion, so we recommend checking results with estimates of variance about regression for fitted models. Fitting growth curves provides a valuable supplement to, and check on computer models of populations.     

Modeling the growth of individuals in crowded plant populations

Aims We present an improved model for the growth of individuals in plant populations experiencing competition.Methods Individuals grow sigmoidally according to the Birch model, which is similar to the more commonly used Richards model, but has the advantage that initial plant growth is always exponential. The individual plant growth models are coupled so that there is a maximum total biomass for the population. The effects of size-asymmetric competition are modeled with a parameter that reflects the size advantage that larger individual have over smaller individuals. We fit the model to data on individual growth in crowded populations of Chenopodium album .Important findings When individual plant growth curves were not coupled, there was a negative or no correlation between initial growth rate and final size, suggesting that competitive interactions were more important in determining final plant size than were plants' initial growth rates. The coupled growth equations fit the data better than individual, uncoupled growth models, even though the number of estimated parameters in the coupled competitive growth model was far fewer, indicating the importance of modeling competition and the degree of size-asymmetric growth explicitly. A quantitative understanding of stand development in terms of the growth of individuals, as altered by competition, is within reach.     

Determination of the parameters of self-association by direct fitting of the omega function

Nonlinear regression is used to fit the omega function vs. protein concentration curves (first described by B.K. Milthorpe, P.D. Jeffrey and L.W. Nichol, Biophys. Chem. 3 (1975) 169) obtained from sedimentation equilibrium experiments on self-associating macromolecules. Nonlinear regression allows the direct fit of these curves with discrete or indefinite self-association reaction models in order to obtain values for the equilibrium constants and second virial coefficient. The method is independent of the choice of reference concentration and avoids the original method of extrapolating an omega function curve to zero concentration and then using the extrapolated value to construct a monomer activity curve used for analysis. This extrapolation can become very difficult for mild to strong self-associations where incorrectly extrapolated values lead to systematic error in the monomer activity curves. The method is applied to results from a mild, indefinite self-association, exemplified by the self-association of human spectrin, and to computer-simulated data of weak, mild and strong, indefinite self-associations.     

十三碳二元酸发酵过程菌体生长期动力学模型及其应用

介绍了由十三碳烷烃生产十三碳二元酸的发酵过程,对其中的菌体生长期的代谢过程进行了分析。提出了以CO₂释放率判断菌体生长状况的方法,据此可确定进入产酸期的最佳时间．建立了菌体生长期底物消耗及菌体生长的动力学模型,对模型参数进行了回归估值。并对菌体生长期进行了拟合。结果表明,模型的计算值和实测值吻合得较好,平均相对偏差为2．4％。利用所建模型对菌体生长期进行多种操作条件下的模拟计算,结果表明,提高蔗糖浓度及初始菌体浓度均能显著地提高菌体生长期结束时的菌体浓度。     

Dictyocaulus viviparus: isolation and characterization of a recombinant antigen with potential for immunodiagnosis.

Crude adult worm antigen of Dictyocaulus viviparus was examined for specific antigens by SDS-PAGE and immunoblotting using sera from cattle experimentally infected with D. viviparus, vaccinated with a normal or a reduced dosage of the commercial lungworm vaccine, and helminth-free cattle. A D. viviparus-specific region M(r) 18,000 was identified and isolated. A lambda ZAP II cDNA expression library consisting of 4.4 x 10(5) recombinant clones (88% of the total number of clones) was constructed from D. viviparus adult worm mRNA. Rabbit antiserum to the M(r) 18,000 antigen was used to screen the cDNA library and eight positive clones were picked and allocated to the same antigenic family by sibling analysis. All clones were subcloned into the plasmid pGEX-2T, and the clone with highest expression yields was expressed as a glutathione S-transferase fusion protein (DvGST3-14) or, after cleavage with thrombin, as pure recombinant parasite protein (Dv3-14). The native parasite antigen encoded by the clone was identified. The immunodiagnostic potential of the recombinant proteins was assessed by immunoblotting.