State vector models of the cell cycle II: The first three moments of the transit time distribution.

A discrete time state vector model (the Hahn model) has been used to simulate many experiments in cell kinetics. In the first paper in this series the authors described a new method to define the parameters of the Hahn model suitable for use in automatic fitting of fraction of labelled mitoses (FLM) experiments. In this paper it is shown how to compute the first three moments of the transit time distribution which arises from a Hahn model. These moments are compared analytically and numerically to the corresponding moments of the distribution the authors used to define the Hahn model. Finally, the problems involved in estimating the moments of the transit time distribution observed in fitting FLM data using a Hahn model are discussed.     

THE LABELLED MITOSES CURVE AND THE ESTIMATION OF THE PARAMETERS OF THE CELL CYCLE

A method is described for fitting a 'fraction labelled mitoses'curve to a set of data points and for estimating the values of the best fitting parameters of the cell cycle. Estimates of the SE of the parameters are obtained. The method depends on the fact that when gamma distributions are used to describe the durations of the phases of the cell cycle, the Laplace transform of a FLM curve can be described by simple analytic functions enabling a least squares fit to be made to a set of Laplace transforms of the experimental data. The method is easy to program and quick to execute.     

State-vector models of the cell cycle 1. Parametrization and fits to labeled mitoses data

In this paper we study the Hahn model of the cell cycle from the point of view that a cell population's age distribution is more relevant to labeled mitoses data than is the distribution of its transit times.Closed-form relationships are derived between the transition probabilities of the Hahn model and the transit time of the mean of a cohort of labeled cells (with the variance of their transit time through mitosis). Constraints result which define the acceptable values for the number of ages in the state vector and the length of the time step (rarely does the dimension of the state vector equal the number of time steps in the generation time).A generalization to distinct probabilities for G₁, S and G₂M is presented, and the automatic fitting of fraction-labeled mitoses (FLM) data is described. The doubling time of the population is used to define the daughter factor, via the largest eigenvalue of the state transition matrix. The performance of the generalized Hahn model is compared to that of other commonly used fitting methods using two sets of FLM data from the literature. The synthesis of continuous labeling curves is discussed as an independent check of the parametrization. Based on the stable age distribution resulting from fits to experimental FLM data, it is shown that a nonlinear relationship exists between biological age and time.     

THE TECHNIQUE OF LABELLED MITOSES: ANALYSIS BY AUTOMATIC CURVE-FITTING

An improved method is described for the analysis of data obtained by the technique of labelled mitoses. It is a development of the method described by Barrett (1966) in which theoretical curves are computed on the basis of a model which assumes that the phases G¹, S and G² are described by independent log-normal distributions; the analysis consists in finding a form of this model which gives a labelled mitoses curve which is the best fit to the available data. This fitting procedure has now been made automatic. No comprehensive indication of the goodness of fit can be given, although in the analysis of over fifty sets of data the method appears to have worked well.
A supplementary computer program is described which, on the basis of three separate assumed modes of cell loss, calculates the form of the age distributions and theoretical continuous labelling curves. This allows growth fraction to be calculated in a way which takes account of the distribution of phase durations and the non-rectangular age distributions of expanding cell populations. It also gives an opportunity to study the implications of continuous labelling data as regards the mode of cell loss.
A comparison is made between the present method of labelled mitoses curve analysis and the empirical rules which have often been used.     

CIRCADIAN INFLUENCE ON THE WAVE FORM OF THE FREQUENCY OF LABELED MITOSES IN MOUSE CORNEAL EPITHELIUM

The FLM was studied in mouse corneal epithelium at two different phases of the mouse circadian system. The following results were obtained: (1) a circadian rhythm was found in the corneal mitotic index of both experimental groups, with the peak near 09.00 hours, the trough near 21.00 hours and a 10-fold difference in values between the peak and the trough; (2) when ³H-TdR was injected at 09.00 hours, the first wave of labelled mitoses rose earlier, attained a broader peak and fell later when compared to the first wave of labeled mitoses obtained when ³H-TdR was injected at 21.00 hours; and (3) in both experimental groups there was a suggestion of a second wave of labeled mitoses, especially in the group injected with ³H-TdR at 21.00 hours, where a flat second wave appeared in the region of 60–90 hr. Transit times derived from FLM curves obtained from such synchronous populations of cells are of questionable validity. Circadian rhythmicity and other causes of synchronization must be ruled out before accepting a single FLM curve as a valid indication of transit times.     

Cell kinetic studies on the JB-1 ascites tumour of the mouse

Abstract. The FLM method, modified by double labelling with [³H]- and [¹⁴C]-thymidine, has been applied to the 4-day old JB-1 ascites tumour of the mouse. It results in well separated waves of purely [³H]- and purely [¹⁴C]-labelled mitoses, which show a remarkable asymmetry with long tails to the right. The following values for the mean transit times of the cells have been derived from this FLM curve, for a tumour age of 4–6 days: T_C= 32.5 hr, T_S= 16.7 hr, T_G1= 3.7 hr, T_G1= 11.0 hr and T_M= 1.1 hr. A further evaluation of the FLM curve, however, is difficult, due to the non-stationary growth of the tumour. A number of other experimental findings (growth curve, decrease of the labelling and mitotic index with increasing tumour age, two single-labelled FLM curves starting 4 and 6 days after tumour inoculation) indicate that the cell cycle time increases during the experimental period of the double-labelled FLM curve (about 2 days). A lengthening of the cycle time should result in an increasing enlargement of the areas under the waves of the modified FLM curve. However, such an increase in area has not been found; the areas are constant. All the results of the present cell kinetic studies would be consistent if it were postulated that the cell cycle time lengthens with increasing tumour age up to about 4 days after inoculation, then remains relatively constant at between 4 and 6 days and thereafter increases again. Short-term double labelling experiments suggest that this is actually the case. Under the assumption of nearly constant phase durations during the 5th and 6th day of tumour growth further conclusions can be drawn from the modified FLM curve. In particular, it follows that the transit times of the cells through successive cycle phases are uncorrelated and the variances of the transit times through a cycle phase are proportional to the duration of this phase.     

GRAIN COUNT THRESHOLD DEPENDENCE OF FLM CURVE PATTERN AND CELL CYCLE PARAMETERS OF HEPATOCYTES IN VIVO

FLM curves from hepatocytes of regenerating rat liver in vivo were compared at different grain count thresholds. Estimates of cell cycle phases derived from curves with thresholds decreasing from 15 to 1 grain (background 0.2 grains per nuclear area) revealed a prolongation of t_s from 6.6 to 9.5 hr, at the expense of t_G2M, and t_G1, whereas t_c remained constant. A similar pattern was observed in FLM curves at various threshold levels for hepatocytes localized in subunits of the liver lobule along the vascular axis from afferent to efferent pole. The shapes of these FLM curves indicated an intralobular gradient of reutilizable labelled material. The use of two different threshold levels is crucial for proper selection of FLM curves to evaluate cell cycle phases in regenerating rat liver: first, a threshold to exclude the autoradiographic background, and a second one to avoid errors due to reutilization of labelled DNA precursors. Each threshold has its own implications for the estimation of cell cycle phases.     

Pharmacological curve fitting to analyze cutaneous adrenergic responses

Although dose-response curves are commonly used to describe in vivo cutaneous α-adrenergic responses, modeling parameters and analyses methods are not consistent across studies. The goal of the present investigation was to compare three analysis methods for in vivo cutaneous vasoconstriction studies using one reference data set. Eight women (22 ± 1 yr, 24 ± 1 kg/m(2)) were instrumented with three cutaneous microdialysis probes for progressive norepinephrine (NE) infusions (1 × 10(-8), 1 × 10(-6), 1 × 10(-5), 1 × 10(-4), and 1 × 10(-3) logM). NE was infused alone, co-infused with NG-monomethyl-l-arginine (l-NMMA, 10 mM) or Ketorolac tromethamine (KETO, 10 mM). For each probe, dose-response curves were generated using three commonly reported analyses methods: 1) nonlinear modeling without data manipulation, 2) nonlinear modeling with data normalization and constraints, and 3) percent change from baseline without modeling. Not all data conformed to sigmoidal dose-response curves using analysis 1, whereas all subjects' curves were modeled using analysis 2. When analyzing only curves that fit the sigmoidal model, NE + KETO induced a leftward shift in ED(50) compared with NE alone with analyses 1 and 2 (F test, P < 0.05) but only tended to shift the response leftward with analysis 3 (repeated-measures ANOVA, P = 0.08). Neither maximal vasoconstrictor capacity (E(max)) in analysis 1 nor %change CVC change from baseline in analysis 3 were altered by blocking agents. In conclusion, although the overall detection of curve shifts and interpretation was similar between the two modeling methods of curve fitting, analysis 2 produced more sigmoidal curves.     

不同性别河北柴鸡早期生长规律及其生长曲线拟合

本研究运用Logistic、Gompertz和Bertalanffy三种非线性模型对不同性别河北柴鸡早期生长规律和生长曲线进行分析及拟合比较.结果表明,3种曲线模型拟合度均达到0.99以上,但Gompertz曲线模型在拟合度和预测极限生长量、拐点周龄和最大周增重等方面相对较好.进一步分析表明,河北柴鸡公鸡的极限体重和拐点体重均高于母鸡,拐点周龄性别间差异不大,公鸡最大周增重与实际观测值接近.本文有助于了解不同性别河北柴鸡各自的生长模式及其对营养、环境的需求,为开展规模化饲养提供参考.     

Estimating polymorphic growth curve sets with nonchronological data

1. When we collect the growth curves of many individuals, orderly variation in the curves is often observed rather than a completely random mixture of various curves. Small individuals may exhibit similar growth curves, but the curves differ from those of large individuals, whereby the curves gradually vary from small to large individuals. It has been recognized that after standardization with the asymptotes, if all the growth curves are the same (anamorphic growth curve set), the growth curve sets can be estimated using nonchronological data; otherwise, that is, if the growth curves are not identical after standardization with the asymptotes (polymorphic growth curve set), this estimation is not feasible. However, because a given set of growth curves determines the variation in the observed data, it may be possible to estimate polymorphic growth curve sets using nonchronological data.
2. In this study, we developed an estimation method by deriving the likelihood function for polymorphic growth curve sets. The method involves simple maximum likelihood estimation. The weighted nonlinear regression and least‐squares method after the log‐transform of the anamorphic growth curve sets were included as special cases.
3. The growth curve sets of the height of cypress (Chamaecyparis obtusa) and larch (Larix kaempferi) trees were estimated. With the model selection process using the AIC and likelihood ratio test, the growth curve set for cypress was found to be polymorphic, whereas that for larch was found to be anamorphic. Improved fitting using the polymorphic model for cypress is due to resolving underdispersion (less dispersion in real data than model prediction).
4. The likelihood function for model estimation depends not only on the distribution type of asymptotes, but the definition of the growth curve set as well. Consideration of these factors may be necessary, even if environmental explanatory variables and random effects are introduced.

     

镇江北固山湿地虉草季节生长动态研究

在虉草(Phalaris arundinacea)整个生长季中,定期测量其株长、鞘高、叶龄及生物量等生长指标,分析和研究虉草的季节生长动态。结果显示:虉草各指标的季节生长动态基本一致,皆呈“S”型曲线,且均以三次方程拟合效果最佳。绝对生长速率和相对生长速率基本同步,呈单峰型曲线,但在负增长出现的时间上稍有差异。各生长指标累积绝对、相对生长速率的季节动态也呈“S”型曲线,同样三次方程拟合效果最佳。     

APOPTOSIS OF EHRLICH MOUSE ASCITES CELLS INDUCED BY LONG-TERM EXPOSURE OF WEAK ELECTROMAGNETIC FIELDS IN VIVO

To study the possibility of apoptosis of tumor cells induced by weak electromagnetic fields (EMFs) in vivo, mice were inoculated with Ehrlich ascites cells and exposed to a long-term electromagnetic field (1 mT, 700 KHz). During the treatment, growth curves of mice were measured and compared between exposed and sham-exposed mice. The results show that the growth curves of healthy controls agree well with the ideal curve of logistic growth, but the growth curves of cancer mice deviate from the ideal curve. There is no difference in growth curves between exposed, and sham-exposed healthy mice, and they both agree with the ideal curve. However, a notable difference in growth curves between exposed and sham-exposed cancer mice was obtained. Moreover, the curves of sham-exposed mice deviate even more than those of the exposed mice; in other words, the growth curves of Ehrlich ascites mice deviate from the ideal curve of healthy mice but are shifted toward it by the EMF treatments. After the treatment, apoptosis of Ehrlich ascites cells from inoculated mice was analyzed by several methods, including flow cytometry, fluorescence microscopy, and DNA gel electrophoresis. Statistical analysis from flow cytometry shows that the apoptotic ratio of cells from exposed Ehrlich ascites mice was significantly higher than that from sham-exposed treated mice. Microscopic observation of Ehrlich ascites cells stained with acridine orange (AO) and propidium iodide (PI) showed typical apoptotic changes in exposed animals whose cell nuclei were highly condensed or fragmented and uniformly stained green by the AO, whereas cell nuclei from sham-exposed mice were stained green and showed a fine reticular pattern. Agarose gel electrophoresis of DNA from exposed mice showed that the chromatin DNA exhibited ladders, a characteristic feature of internucleosomal degradation of DNA by EMF treatments. For interactions between external electromagnetic fields and DNA, the mechanism of apoptosis of tumor cells induced by weak EMFs is discussed.     

An autoradiographical topological study of the epithelial cell proliferative activity in the ascending colon of the mouse using a modified technique for determining the fraction of labelled mitoses (FLM)

In a previous study of the mouse ascending colon we showed that epithelial cell proliferation is more active on the tops of mucosal folds than in the flat mucosal areas between folds. In order to define the cytokinetic characteristics of this phenomenon more precisely, we have undertaken an autoradiographic investigation with 3H-thymidine. The fraction of labelled mitoses (FLM) curves and cell labelling indices have been worked out in fold top areas (FTA) and on flat areas off the folds (OFA). A new method has been adopted to draw the FLM curves. The results show no difference between the cell cycle times or in the times corresponding to G2-phase and S-phase in FTA and OFA. In contrast, the proliferative cell compartment in FTA is 40% higher than in OFA.     

Mlab--a mathematical modeling tool

An interactive interpreter called Mlab is described. One uses Mlab by typing commands. In this sense, Mlab is a programming language. It has various mathematical and graphical facilities which make it a useful tool for mathematical modeling. The curve fitting capabilities of Mlab are augmented with differential-equation-handling and matrix-manipulation capabilities which provide a powerful and civilized facility for curve fitting. Many people are engaged in this activity, and, in general, they use programs which are neither sufficiently general nor easy to use. (Some conventional programming is usually required, for example.) Mlab purports to be easier than alternate approaches. The nature of Mlab is discussed with accompanying examples. The main example is the use of curve fitting to determine molecular weight from ultracentrifuge data. This example was chosen because it exhibits a special feature of Mlab, namely the root operator, which appears in the definition of the model function.     

A SIMPLIFIED METHOD OF DNA DISTRIBUTION ANALYSIS

Current methods of analysing DNA distributions utilize complex mathematical expressions that require the use of large non-linear curve fitting methods and, consequently, large computers. This paper presents a new method of analysing DNA distributions of asynchronously growing or mildly perturbed cells. The S phase fraction is obtained by fitting a second degree polynomial to that part of the distribution, mid S phase, which is not influenced by either the G₁ or the G₂+ M peaks. The method is simple and fast, it exceeds the accuracy of other methods and it can be used on a large desk calculator or mini-computer.     

Assessment of Shape Changes of Mistletoe Berries: A New Software Approach to Automatize the Parameterization of Path Curve Shaped Contours

Photographs of mistletoe (Viscum album L.) berries taken by a permanently fixed camera during their development in autumn were subjected to an outline shape analysis by fitting path curves using a mathematical algorithm from projective geometry. During growth and maturation processes the shape of mistletoe berries can be described by a set of such path curves, making it possible to extract changes of shape using one parameter called Lambda. Lambda describes the outline shape of a path curve. Here we present methods and software to capture and measure these changes of form over time. The present paper describes the software used to automatize a number of tasks including contour recognition, optimization of fitting the contour via hill-climbing, derivation of the path curves, computation of Lambda and blinding the pictures for the operator. The validity of the program is demonstrated by results from three independent measurements showing circadian rhythm in mistletoe berries. The program is available as open source and will be applied in a project to analyze the chronobiology of shape in mistletoe berries and the buds of their host trees.     

The labelled mitosis curve for a population consisting of fast and slowly cycling cells

In studies of tumour growth, and particularly of tumour treatment with phase-specific chemotherapeutic agents, the fraction labelled mitosis technique is frequently used to estimate the kinetic properties of the cell population making up the tumour. We show here that the FLM technique is in principle very insensitive to the behaviour of slowly cycling cells, even if these constitute a large proportion of the total cell population. Furthermore, since the rate of DNA synthesis is frequently lower in slowly growing cells than in those growing rapidly, there is a higher probability of labelling error associated with the former cells. In view of these theoretical and experimental considerations, it is suggested that considerable caution be used when applying the FLM technique to heterogeneous cell populations such as those of solid tumours.