Growth of cell populations with arbitrarily distributed cycle durations. I. basic model

A discrete model is proposed describing the growth of cell populations with arbitrary frequency distributions of cycle durations. The model assumes that each cell divides into two cells at the end of its cycle, and that each new cell is assigned an individual cycle duration according to a probability distribution that can be arbitrarily defined. The increase in the cell number is calculated, either from the numbers of cells at earlier time points or from the initial conditions of the population, by a recurrence formula; it is also approximated by the optimal exponential function, whose parameters are determined by the initial conditions. The appropriate average cycle duration is shown not to be the arithmetic or geometric mean, but rather the solution to a more complex equation. Age distributions are calculated and compared with those found in the literature. The results of the model calculations are compared with computer simulations and with observed data on populations of the ciliate Tetrahymena geleii.     

A mathematical analysis of focal adhesion lifetimes and their effect on cell motility

By analyzing the distributions of focal adhesion (FA) lifetimes from different cell types, we found that a gamma distribution best matched the experimental distributions. In all but one case, it was a unimodal, non-symmetric gamma distribution. We used a mathematical model of cell motion to help understand the mechanics and data behind the FA lifetime distributions. The model uses a detach-rate function to determine how long an FA will persist before it detaches. The detach-rate function that produced distributions with a best-fit gamma curve that closely matched that of the data was both force and time dependent. Using the data gathered from the matching simulations, we calculated both the cell speed and mean FA lifetime and compared them. Where available, we also compared this relationship to that of the experimental data and found that the simulation reasonably matches it in most cases. In both the simulations and experimental data, the cell speed and mean FA lifetime are related, with longer mean lifetimes being indicative of slower speeds. We suspect that one of the main predictors of cell speed for migrating cells is the distribution of the FA lifetimes.     

Modelling cell population growth with applications to cancer therapy in human tumour cell lines

In this paper we present an overview of the work undertaken to model a population of cells and the effects of cancer therapy. We began with a theoretical one compartment size structured cell population model and investigated its asymptotic steady size distributions (SSDs) (On a cell growth model for plankton, MMB JIMA 21 (2004) 49). However these size distributions are not similar to the DNA (size) distributions obtained experimentally via the flow cytometric analysis of human tumour cell lines (data obtained from the Auckland Cancer Society Research Centre, New Zealand). In our one compartment model, size was a generic term, but in order to obtain realistic steady size distributions we chose size to be DNA content and devised a multi-compartment mathematical model for the cell division cycle where each compartment corresponds to a distinct phase of the cell cycle (J. Math. Biol. 47 (2003) 295). We then incorporated another compartment describing the possible induction of apoptosis (cell death) from mitosis phase (Modelling cell death in human tumour cell lines exposed to anticancer drug paclitaxel, J. Math. Biol. 2004, in press). This enabled us to compare our model to flow cytometric data of a melanoma cell line where the anticancer drug, paclitaxel, had been added. The model gives a dynamic picture of the effects of paclitaxel on the cell cycle. We hope to use the model to describe the effects of other cancer therapies on a number of different cell lines.     

Plankton size distributions and trophic relations before and after the construction of the storm-surge barrier in the Oosterschelde estuary

Phytoplankton and zooplankton biomass distributions were calculated on a carbon basis for the inland part of the Oosterschelde, in the period before (1983), during (1984, 1986) and after (1987, 1988) the construction of the storm- surge barrier. In all years studied, both phytoplankton and zooplankton distributions are very irregular, and little consistent patterns emerge. The data were used to test the model of Sheldon et al. (1977). The observed standing stock ratios of zooplankton to phytoplankton agree with the model predictions in 1983, and are slightly higher during the period 1984–1987. In 1988, the model predictions are very different from the observed values, because of important changes in the zooplankton species abundance occurring in this year.     

Symmetric Bernoulli Distributions and Generalized Binomial Distributions

The generalized binomial distribution is defined as the distribution of a sum of symmetrically distributed Bernoulli random variates. Several two-parameter families of generalized binomial distributions have received attention in the literature, including the Polya urn model, the correlated binomial model and the latent variable model. Some properties and limitations of the three distributions are described. An algorithm for maximum likelihood estimation for two-parameter generalized binomial distributions is proposed. The Polya urn model and the latent variable model were found to provide good fits to sub-binomial data given by Parkes. An extension of the latent variable model to incorporate heterogeneous response probabilities is discussed.     

Determination of total and active immobilized enzyme distribution in porous solid supports

The total and active immobilized enzyme (IME) distributions in porous supports are studied both theoretically and experimentally. In order to determine experimentally the enzyme distribution profiles within a single particle, we construct a diffusion cell containing controlled-pore glass particles such that the cell would mimic a large pellet support. Our purpose is to study the interplay between the diffusion process within the interparticle void space and immobilization process in the controlled-pore glass particles onto the evolution of the (total and active) enzyme distributions. A mathematical model is developed to describe the interaction of various processes within the diffusion cell. The immobilized enzymes are determined for a system of trypsin and controlled-pore glass particles. The total amount of enzymes are determined by the amino acid analysis, and the active fraction is obtained by an active-site titration. The experimentally measured total IME profiles compare very well with that predicted by the model. The determined active enzyme profile is found to be nonuniform one, and it represents about 40% of the total enzyme immobilized in the support particles.     

Multimodal interval distributions

Summary A mathematical model is presented that is supposed to describe certain types of multimodal interval distributions of neuronal discharges. Basically it consits of the selective interaction between an excitatory and an inhibitory impulse sequence. The theoretical results are compared with nerve cell interval distributions reported in the literature, and with distributions from simulation studies. A possible relationship between the properties of this model and longtailed interval distributions is indicated. Several extensions are discussed.     

Subject-specific modeling of the scapula bone tissue adaptation

Adaptation of the scapula bone tissue to mechanical loading is simulated in the current study using a subject-specific three-dimensional finite element model of an intact cadaveric scapula. The loads experienced by the scapula during different types of movements are determined using a subject-specific large-scale musculoskeletal model of the shoulder joint. The obtained density distributions are compared with the CT-measured density distribution of the same scapula. Furthermore, it is assumed that the CT-measured density distribution can be estimated as a weighted linear combination of the density distributions calculated for different loads experienced during daily life. An optimization algorithm is used to determine the weighting factors of fourteen different loads such that the difference between the weighted linear combination of the calculated density distributions and the CT-measured density is minimal. It is shown that the weighted linear combination of the calculated densities matches the CT-measured density distribution better than every one of the density distributions calculated for individual movements. The weighting factors of nine out of fourteen loads were estimated to be zero or very close to zero. The five loads that had larger weighting factors were associated with either one of the following categories: (1) small-load small-angle abduction or flexion movements that occur frequently during our daily lives or (2) large-load large-angle abduction or flexion movements that occur infrequently during our daily lives.     

An index to compare geographical distributions of species

We present an index for the dissimilarity/distance between geographical distributions based on reporting rates recorded on a regular lattice. Reporting rate data are common, for example, in bird atlas projects where observers fill in check lists of encountered species in a particular area. Our index is a variation of the Euclidean distance, with the contribution of each grid cell weighted by the number of checklists collected for the grid cell, and a scaling factor to ensure that the dissimilarity ranges between zero and one. Reporting rates were transformed to ordered percentile classes to make species with different mean reporting rates comparable. The index was developed for the comparison of distributions of The Atlas of Southern African Birds . We illustrate the dissimilarity index comparing distributions of whydahs and indigobirds (widowfinches), which are specialized brood parasites, to the distributions of their hosts: waxbills and other finches.     

Analysis of particle strain in stirred bioreactors by drop breakage investigations

Understanding of particle strain and drop breakage is relevant for various technical applications. To analyze it, single drop experiments in a breakage cell and evolving drop size distributions in an agitated system are studied. The mechanisms for particle strain and drop breakage are assumed to be comparable for the investigated turbulent flow regime. The agitation process is simulated using a population balance model. This model provides transient prediction capacities at different scales and can be used for scale-up/down projects. The number and the size distributions of daughter fragments for single drops have been studied. The results clearly support the assumption of binary breakage. The most common assumption of a Gaussian distribution for the daughter drop size distribution could not be supported. The evolution of a breakage-dominated toluene/water system was then simulated using different daughter drop size distributions from literature. The computational results were compared with experimental values. All simulations were able to predict the transient Sauter mean diameter excellently but varied strongly in the results on the shape of the distribution. In agreement with the experimental single drop results, the use of a bimodal or a very broad bell-shaped distribution of the daughter drops is proposed for the simulations. Although these results were obtained in a particular vessel for a specific phase system, it can be applied to simulate transient multiphase systems at different scales. We would expect that the general trends observed in this study are comparable to various applications in multiphase bioreactors.     

Estimating Microtubule Distributions from 2D Immunofluorescence Microscopy Images Reveals Differences among Human Cultured Cell Lines

Microtubules are filamentous structures that are involved in several important cellular processes, including cell division, cellular structure and mechanics, and intracellular transportation. Little is known about potential differences in microtubule distributions within and across cell lines. Here we describe a method to estimate information pertaining to 3D microtubule distributions from 2D fluorescence images. Our method allows for quantitative comparisons of microtubule distribution parameters (number of microtubules, mean length) between different cell lines. Among eleven cell lines compared, some showed differences that could be accounted for by differences in the total amount of tubulin per cell while others showed statistically significant differences in the balance between number and length of microtubules. We also observed that some cell lines that visually appear different in their microtubule distributions are quite similar when the model parameters are considered. The method is expected to be generally useful for comparing microtubule distributions between cell lines and for a given cell line after various perturbations. The results are also expected to enable analysis of the differences in gene expression underlying the observed differences in microtubule distributions among cell types.     

Statistical evidence for two major proteins in freeze-fractured gastric parietal cell tubulovesicles and canaliculus

In a previous work, resting and acid-secreting rabbit gastric mucosa were freeze-fractured and shadowed at 45° with Pt-C. The shadow widths of proteic particles of tubulovesicle and canaliculus membranes were measured and compared. It was concluded that the frequency distributions of widths are significantly different in resting and secreting membranes and that each distribution accounts for several subpopulations of homogenous particles. In the present study, an attempt is made to describe the experimental distributions as a mixture of those of two major proteins, say A and B and their aggregates (AA, AB and BB). The modelling, although simple, gave a very satisfactory statistical fit between observed and computed distributions. The comparison of parameters calculated from histamine and ranitidine experimental data further improves the fits and finally, component A accounts for 69% of the particles. Most replica of A particles are heart-shaped and the median shadow widths are 6.1 and 6.8 nm in canaliculus and tubulovesicles respectively. The component B accounts for 31% of the particles. They mainly appear as small barrels and the median shadow widths are 8.8 and 10.3 nm in canaliculus and tubulovesicles respectively. According to calculated parameters and observed particle replica, the onset of secretion does not change the relative ratio of proteins but changes their shapes. Component A should be the (H⁺, K⁺)ATPase whereas debate on the identity of B is wide open.     

西北干旱区10种荒漠植物地理分布与大气候的关系及其可能潜在分布区的估测

 选择面积广阔、地理位置特殊、气候干旱、地形地貌多样、生态系统脆弱以及人类活动长期干扰,而且目前面临着西部大开发和环境保护双重矛盾的西北干旱区作为研究区域,通过收集西北干旱区多种优势种和常见种的地理分布资料,选取10 种分布范围相对明确、资料相对齐备的荒漠植物种：短叶假木贼（Anabasis brevifolia）、木蓼（Atraphaxis frutescens）、沙拐枣（Calligonum mongolicum）、膜果麻黄（Ephedra przewalskii）、裸果木（Gymnocarpos przewalskii）、梭梭柴（Haloxylon ammodendron）、白梭梭（Haloxylon persicum）、尖叶盐爪爪（Kalidium cuspidatum）、松叶猪毛菜（Salsola laricifolia）和合头草（Sympegma regelii）,定量分析其地理分布与气候因子的关系,并据此估测其潜在中心分布区和潜在最大分布范围,与实际分布范围进行比较。结果表明,10种荒漠植物的Holdridge生物温度、降水量和可能蒸散率的平均值、标准差、最大值和最小值比较准确地反映了我国荒漠地区的典型气候特征,说明植物种分布与气候的定量对应关系较好。其可能潜在分布的估测中,中心分布估测图与植物种的实际分布范围有较好的对应性,而在最大可能分布的估测图中,与实际有一定误差及存在一些不合理的地方,但种与种之间有差异。究其原因,首先在于目前所拥有的植物种分布和气象资料的限制,而且植物种的实际分布范围受到人为活动的影响较大,所以无法完全把握植物种与气候的准确定量关系;其次,气候极值所预测的只是植物种分布的最大可能性,而不能表现出植物种现实分布的主要地区,气候均值与标准差所反映的中心分布区,其准确性更可靠;再者,模拟预测仅考虑了植物种与大气候的关系,而没有考虑决定植物分布的其它环境因素,如土壤、基质、地形等地理因素。     

广义模型及分类回归树在物种分布模拟中的应用与比较

比较3个应用较广的模拟物种地理分布模型:广义线性模型(GLM)、广义加法模型(GAM)与分类回归树(CART)对中国树种地理分布模拟的优劣,以提出更为合适的模拟物种地理分布模型,并用于预测气候变化对物种地理分布的影响。3个模型对中国15种树种地理分布的模拟研究表明:除对油松、辽东栎分布的模拟精度稍差外,对其余树种分布的模拟精度均较高,其中以GAM模型最好。结合地理信息系统(GIS),比较分析了这3个模型对青冈、木荷、红松和油松4种树种的地理分布模拟效果,结果亦表明:这3个模型均能很好模拟青冈和木荷的地理分布,而GLM模型对红松分布的模拟结果不太理想,3个模型对油松分布的模拟结果均不甚理想,其中以GLM模型最差。基于3个模型对未来气候变化下青冈与蒙古栎地理分布的预测表明:GLM模型与GAM模型对青冈分布的预测结果较为接近,青冈在未来气候变化情景下向西和向北扩展,而CART模型预测青冈在未来气候变化情景下除有向西、向北扩展趋势外,广东和广西南部的青冈分布区将消失;3个模型均预测蒙古栎在未来气候变化情景下向西扩展,扩展面积的大小为:模型的模拟面积>模型>模型。     

Maxent模型复杂度对物种潜在分布区预测的影响

生态位模型在入侵生物学和保护生物学中具有广泛的应用, 其中Maxent模型最为流行, 被越来越多地应用在预测物种的现实分布和潜在分布的研究中。在Maxent模型中, 多数研究者采用默认参数来构建模型, 这些默认参数源自早期对266个物种的测试, 以预测物种的现实分布为目的。近期研究发现, Maxent模型采用复杂机械学习算法, 对采样偏差敏感, 易产生过度拟合, 模型转移能力仅在低阈值情况下较好。基于默认参数的Maxent模型不仅预测结果不可靠, 而且有时很难解释。在本研究中, 作者以入侵害虫茶翅蝽(Halyomorpha halys)为例, 采用经典模型构建方案(即构建本土模型然后将其转移至入侵地来评估), 利用ENMeval数据包来调整本土Maxent模型调控倍频和特征组合参数, 分析各种参数条件下模型的复杂度, 然后选取最低复杂度的模型参数(即为最优模型), 综合比较默认参数和调整参数后Maxent模型的响应曲线和预测结果, 探讨Maxent模型复杂度对预测结果的影响及Maxent模型构建时所需注意事项, 以期对物种潜在分布进行合理的预测, 促进Maxent模型在我国的合理运用和发展。作者认为, 环境变量的选择至关重要, 需要综合分析其对所模拟物种分布的限制作用和环境变量之间的空间相关性。构建Maxent模型前需对物种分布采样偏差及模型的构建区域进行合理地判断, 模型构建时需要比较不同参数下模型的预测结果和响应曲线, 选取复杂度较低的模型参数来最终建模。在茶翅蝽的分析中, Maxent模型的默认参数和最优模型参数不同, 与Maxent模型默认参数相比, 采用调整参数后所构建的模型预测效果较好, 响应曲线较为平滑, 模型转移能力较高, 能够较为合理反映物种对环境因子的响应和准确地模拟该物种的潜在分布。     

A numerical model for temperature distribution and thermal damage calculations in teeth exposed to a CO2 laser

A numerical axisymmetrical model which may be used for the evaluation of laser dental treatments is presented. This model facilitates the calculations of the temperature distribution and of thermal damage to pulp tissue caused by CO₂ laser irradiation. Temperature distributions are compared with analytical, experimental, and numerical results presented in the literature. The conformity obtained is good. It is shown that this model can be used over a wider range of time intervals and physical conditions than a previous numerical model. In addition, thermal damage is calculated for the temperature distributions presented in this paper. This model can be utilized for the optimization of exposure parameters to minimize pulp damage in the application of lasers for dental treatment.     

Molecular Dynamics Investigation of Bond Ordering of Unsaturated Lipids in Monolayers

Molecular dynamics simulations of three model lipid monolayers of 2,3-diacyl-D-glycerolipids, that contained stearoyl (18:0) in the position 3 and oleoyl (18:9cis), linoleoyl (18:26cis), or linolenoyl (18:33cis) in the position 2, have been carried out. The simulation systems consisted of 24 lipid molecules arranged in a rectangular simulation cell, with periodic boundary conditions in the surface plane. 1 nanosecond simulations were performed at T = 295 K. C-C and C-H bond order parameter profiles and the bond orientation distributions about the monolayer normal have been calculated. The relation of the distributions to the order parameters was analyzed in terms of maxima and widths of the distributions. The cis double bond order parameter is found to be higher than those of adjacent single C-C bonds. The widths of the two distributions of C-H bonds of the cis double bond segment in di- and triunsaturated molecules are much smaller than that obtained for methylene group located between the double bonds. The bond orientation distribution function widths depend on both the segment location in the chain and the segment chemical structure.     

A validated model of GAG deposition,cell distribution,and growth of tissue engineered cartilage cultured in a rotating bioreactor

In this work a new phenomenological model of growth of cartilage tissue cultured in a rotating bioreactor is developed. It represents an advancement of a previously derived model of deposition of glycosaminoglycan (GAG) in engineered cartilage by (i) introduction of physiological mechanisms of proteoglycan accumulation in the extracellular matrix (ECM) as well as by correlating (ii) local cell densities and (iii) tissue growth to the ECM composition. In particular, previously established predictions and correlations of local oxygen concentrations and GAG synthesis rates are extended to distinguish cell secreted proteoglycan monomers free to diffuse in cell surroundings and outside from the engineered construct, from large aggrecan molecules, which are constrained within the ECM and practically immovable. The model includes kinetics of aggregation, that is, transformation of mobile GAG species into immobile aggregates as well as maintenance of the normal ECM composition after the physiological GAG concentration is reached by incorporation of a product inhibition term. The model also includes mechanisms of the temporal evolution of cell density distributions and tissue growth under in vitro conditions. After a short initial proliferation phase the total cell number in the construct remains constant, but the local cell distribution is leveled out by GAG accumulation and repulsion due to negative molecular charges. Furthermore, strong repulsive forces result in expansion of the local tissue elements observed macroscopically as tissue growth (i.e., construct enlargement). The model is validated by comparison with experimental data of (i) GAG distribution and leakage, (ii) spatial‐temporal distributions of cells, and (iii) tissue growth reported in previous works. Validation of the model predictive capability—against a selection of measured data that were not used to construct the model—suggests that the model successfully describes the interplay of several simultaneous processes carried out during in vitro cartilage tissue regeneration and indicates that this approach could also be attractive for application in other tissue engineering systems. Biotechnol. Bioeng. 2010. 105: 842–853. © 2009 Wiley Periodicals, Inc.     

A self-similarity model for the occupancy frequency distribution

The shapes of interspecific range-size distributions at scales finer than the geographic range are highly variable. However, no numerical model has been developed as a basis for understanding this variation. Using self-similarity conditions, we present an occupancy probability transition (OPT) model to investigate the effect of sampling scale (i.e. sample grain) and species saturation (strongly positively correlated with the fractal dimension) on the shape of occupancy frequency distributions (fine scale expression of range-size distributions). In accordance with empirical observations, the model showed that core-modes are likely to be rare in occupancy frequency distributions. The modal occupancy shifted from core to satellite with an increase in sample grain (from coarse scale to fine scale) at a linear rate after log-transformation of occupancy. Saturation coefficients above a particular threshold generated multimodality. Bimodal distributions arose from a combination of different occupancy probability distributions (OPDs), with species-specific saturation coefficients generating occupancy frequency distributions of the shape commonly observed empirically, i.e. bimodal with a dominant satellite mode. This is a consequence of the statistical properties of the OPD, and is also largely insensitive to species richness. The OPT model thus provides a null model for the shape of occupancy frequency distributions. Furthermore, it demonstrates that the sample grain of a study, sampling adequacy (based on a linear sampling assumption) and the distribution of species saturation coefficients in a community are together largely able to explain the patterns observed in empirical occupancy frequency distributions.