Analysis of DNA synthesis rate of cultured cells from flow cytometric data

The rate of DNA synthesis along S phase is estimated from flow cytometric histograms on the basis of a mathematical model of a cell population. In the absence of loss, the model expresses the population kinetics in terms of DNA synthesis rate, S-phase influx, and population size. A single histogram is sufficient to determine the DNA synthesis rate when the population is in balanced exponential growth. Two suitably chosen histograms are necessary if the S-phase influx is exponential in a time interval longer than the S-phase duration. The analysis procedure was tested on published autoradiographic data and applied to three cultured cell lines (CM-S, 3LL, and M14 cells) that show various patterns of DNA distribution. In each case the cell-cycle fractions, the DNA synthesis rate, and the S-phase duration were obtained.     

Error analysis in flow microfluorometry

Sources of error in a typical algorithm for the analysis of single flow-microfluorometric histograms are identified. A new statistical model for such data is presented, by means of which the error sources are quantitatively investigated. These theoretical investigations lead to three practical observations: A more detailed characterization of the fluorescence dispersion process is needed for a more refined algorithm. Levels of dispersion typically experienced are such that from a single histogram the distribution of cells within S-phase cannot be finely resolved; but the crude distribution of cells among the three phases G1, S, and G2-M may be accurately estimated. If currently typical levels of dispersion can be halved, then the S-phase distribution can be finely resolved.     

Error analysis in flow microfluorometry

Sources of error in a typical algorithm for the analysis of single flow-microfluorometric histograms are identified. A new statistical model for such data is presented, by means of which the error sources are quantitatively investigated. These theoretical investigations lead to three practical observations: A more detailed characterization of the fluorescence dispersion process is needed for a more refined algorithm. Levels of dispersion typically experienced are such that from a single histogram the distribution of cells within S-phase cannot be finely resolved; but the crude distribution of cells among the three phases G₁, S, and G₂-M may be accurately estimated. If currently typical levels of dispersion can be halved, then the S-phase distribution can be finely resolved.     

Statistical evaluation of cell kinetic data from DNA flow cytometry (FCM) by the EM algorithm

Flow cytometric DNA measurements yield the amount of DNA for each of a large number of cells. A DNA histogram normally consists of a mixture of one or more constellations of G0/G1-, S-, G2/M-phase cells, together with internal standards, debris, background noise, and one or more populations of clumped cells. We have modelled typical DNA histograms as a mixed distribution with Gaussian densities for the G0/G1 and G2/M phases, an S-phase density, assumed to be uniform between the G0/G1 and G2/M peaks, observed with a Gaussian error, and with Gaussian densities for standards of chicken and trout red blood cells. The debris is modelled as a truncated exponential distribution, and we also have included a uniform background noise distribution over the whole observation interval. We have explored a new approach for maximum-likelihood analyses of complex DNA histograms by the application of the EM algorithm. This algorithm was used for four observed DNA histograms of varying complexity. Our results show that the algorithm works very well, and it converges to reasonable values for all parameters. In simulations from the estimated models, we have investigated bias, variance, and correlations of the estimates.     

DNA synthesis rate and unlabeled cells with S-phase DNA content in P388 leukemic cells in vivo studied by flow cytometry and cell sorting

DNA synthesis kinetics of P388 leukemic cells growing in ascites form in BDF1 hybrid mice were investigated during the periods of exponential growth and growth restriction. Incorporation of tritiated thymidine, and in some instances tritiated uridine, was studied by autoradiography in cells sorted from S-phase fractions during DNA flow cytometry. During exponential growth continuous labeling with tritiated thymidine indicated a growth fraction of unity, whereas the growth fraction was about 30% during growth restriction. At this growth phase the majority of cells with S phase DNA content remained unlabeled after pulse labeling with tritiated thymidine or uridine, indicating that both the "salvage" and the "de novo" DNA synthesis pathways were blocked in most S-phase cells. After pulse labeling with tritiated thymidine the DNA synthesis rate pattern was investigated by sorting of consecutive fractions of cells throughout the S phase followed by quantitative autoradiography. With exception of a reduced rate in the middle of S phase, the DNA synthesis rate increased as the cells progressed through S phase during exponential growth. In contrast, the DNA synthesis rate pattern had a relative peak in the middle of S phase during growth restriction, which is otherwise characterized by a low mean DNA synthesis rate.     

DNA histogram debris theory and compensation.

In this paper we describe a theory of DNA histogram debris generation and compensation that can be applied to paraffin-embedded frozen tissue preparations. The theory predicts the distribution of fragments generated from single and multiple random sectioning of three-dimensional ellipsoids representing nuclei. The fragment distribution is assumed to be a major component of the underlying debris in DNA histograms. A comparison of S-phase fractions (SPF) from matched tissue prepared by frozen and formalin-fixed paraffin-embedded DNA methods demonstrates the usefulness of the theory.     

Reliability of DNA cytometric S-phase analysis in surgical biopsies: assessment of systematic and sampling errors and comparison between results obtained by image and flow cytometry

Three major parameters in DNA histograms that contribute to the reliability of S-phase analysis were evaluated. These parameters are (1) the extent of background in relation to the amount of S-phase cells (and the validity of its subtraction), (2) the size of the "free" S-phase range (S(free)), and (3) the sampling error of cell counting. Tests in histograms obtained from surgical biopsies by flow cytometry (FCM) showed that the background subtraction is reliable if the found S-phase fraction is higher than the fraction of background events in the histogram range of the cell population. The size of S(free) was determined in computer-generated test histograms as a function of variables such as the coefficient of variation (CV) and the DNA index (DI). To calculate the sampling error of cell counting above background and in S(free), a model was developed that was validated by experimental data. This error can serve as an indicator of the uncertainty in S-phase analysis. The poor correlation found between %S values measured by image cytometry (ICM) and FCM in surgical biopsies was assigned to high uncertainty by low cell numbers in ICM histograms. A method is proposed to estimate quantitatively the reliability of S-phase analysis that can facilitate the interpretation of results.     

Macromolecule synthesis of Escherichia coli BB at a lower or transient growth state.

Macromolecule synthesis in Escherichia coli BB at lower growth rates was investigated. The results indicate that a deviation in ribonucleic acid (RNA) content per cell at a lower growth rate from the exponential relationship to a specific growth rate is entirely attributable to the presence of nonviable cells, in which the RNA content is lower than in viable cells. Based on this fact, a mathematical expression of macromolecule contents versus specific growth rate was devised. Moreover, continuous changes in macromolecule content during unbalanced growth from late-logarithmic phase to stationary phase were measured. Although growth rates changed continuously, the data on deoxyribonucleic acid (DNA) or RNA content versus the specific growth rate calculated from the increments in cell number satisfactorily fitted the exponential lines obtained under balanced growth at a higher growth rate. However, no such relationship was observed in the plot of DNA or RNA content versus the specific growth rate calculated from the increments in optical density.     

Macromolecule synthesis of Escherichia coli BB at a lower or transient growth state.

Macromolecule synthesis in Escherichia coli BB at lower growth rates was investigated. The results indicate that a deviation in ribonucleic acid (RNA) content per cell at a lower growth rate from the exponential relationship to a specific growth rate is entirely attributable to the presence of nonviable cells, in which the RNA content is lower than in viable cells. Based on this fact, a mathematical expression of macromolecule contents versus specific growth rate was devised. Moreover, continuous changes in macromolecule content during unbalanced growth from late-logarithmic phase to stationary phase were measured. Although growth rates changed continuously, the data on deoxyribonucleic acid (DNA) or RNA content versus the specific growth rate calculated from the increments in cell number satisfactorily fitted the exponential lines obtained under balanced growth at a higher growth rate. However, no such relationship was observed in the plot of DNA or RNA content versus the specific growth rate calculated from the increments in optical density.     

Blocking kinetics of Cl- channels in colonic carcinoma cells (HT29) as revealed by 5-nitro-2-(3-phenylpropylamino)benzoic acid (NPPB)

The blocking effect of 5-nitro-2-(3-phenylpropylamino)benzoic acid (NPPB) was investigated on single Cl- channels of the cultured human colon carcinoma cells, HT29. In the absence of NPPB, the open-time histogram yielded two time constants, with 0.9 ms and 33 ms, whereas the closed-time distribution could be fitted by a single exponential with a time constant of 0.7 ms. Addition of NPPB in the range 1-50 microM induced brief closing events of the single-channel current. This resulted in a decrease of the long open-time constant to 2.1 ms and in an increase of the closed-time constant to 1.8 ms at 50 microM NPPB concentration. The short open-time constant did not change at low blocker concentration (1 microM), but could no longer be resolved at higher concentrations. The open-state probability decreased from 0.9 (control conditions) to 0.5 at 50 microM NPPB. The Hill plot yielded a Hill coefficient of about 0.7, compatible with one NPPB molecule inhibiting one channel molecule. The kinetics of channel gating are described by a sequential model with one closed and two open states. Since in the presence of NPPB no additional time constant appeared in the time histograms, we assumed the same kinetic scheme as under control conditions, and hypothesize that NPPB has an influence on rate constants.     

Kinetics of protein and DNA synthesis studied by mathematical modelling of flow cytometric protein and DNA histograms

Abstract. Mathematical models for histograms of cellular protein content as measured by flow cytometry were developed, based on theoretical protein distributions. These were derived from the age distribution of cells and the accumulation function for cellular protein content as a function of age within the cell cycle. A model assuming an exponential age distribution and an exponential protein. accumulation function was found to give the best representation of protein histograms of exponentially growing NHIK 3025 cells. This is in good agreement with the known kinetic behaviour of such cells. By the combined use of the protein histogram model and a similar model for DNA content, and assuming linear DNA accumulation during S, the fraction of cells in S, as a function of cellular protein content, was simulated. This function showed good agreement with values of the [³H]TdR labelling index scored in cells sorted by flow cytometry from 5-channel intervals of the protein histogram. The protein and DNA histogram models were combined into a two-dimensional model for correlated protein/DNA measurements. Comparison between simulated data and experimentally derived two-dimensional protein/DNA histograms gave further support to the cell kinetic assumptions underlying the models, but also identified some minor deviations which could not be recognized in the analysis of the one-dimensional histograms.     

Selection of optimal model for the DNA histogram by analysis of error of estimated parameters.

The ability of four different mathematical models of the DNA histogram to give accurate estimates for the fractions of cells in G1, S, and G2 + M has been investigated. The models studied differ in the form and number of parameters of the function used to represent cells in S-phase. Results obtained from simulated DNA histograms suggest that the standard deviations of the model parameters increase exponentially with the width of the G1 and G2 + M peaks of the histogram. Error analysis is presented as a method to select a model of optimal complexity in relation to the resolution provided by the data in a given set of DNA histograms. Introduction of additional parameters improves the agreement between model and data but may result in a less well-posed model. A model with an optimal number of parameters can therefore be found that will yield parameter estimates with the smallest possible standard deviations.     

On microbial states of growth†

It is crucial to the reproducibility of results and their proper interpretation that the conditions under which experiments are carried out be defined with rigour and consistency, in this review we attempt to clarify the differences and interrelationships among steady, balanced and exponential states of culture growth. Basic thermodynamic concepts are used to introduce the idea of steady-state growth in open, biological systems. The classical, sometimes conflicting, definitions of steady-state and balanced growth are presented, and a consistent terminology is proposed. The conditions under which a culture in balanced growth is also in exponential growth and in steady-state growth are indicated. It is pointed out that steady-state growth always implies both balanced and exponential growth, and examples in which the converse does not hold are described. More complex situations are then characterized and the terminology extended accordingly. This leads to the notion of normal growth and growth that can be synchronous or otherwise unbalanced but still reproducible, and to the condition of approximate steady state manifested by growth in batch culture and by asymmetrically dividing cells, which is analysed in some detail.     

Quantitative aspects of cellular turnover

Living organisms do not just grow by synthesizing cellular components. As part of the necessary steps for existence, some components are degraded after synthesis. Even for bacteria in balanced, exponential growth some substances, under some conditions, are turned over. In other phases of growth turnover can be much more extensive, but it is still selective. This review covers studies with animals as a way to put the studies on microorganisms in perspective. The history, the mathematics, and experimental design of turnover experiments are reviewed. The important conclusion is that most of the proteins during balanced growth are very stable in bacteria, although ribosomal proteins are degraded under starvation conditions. Another generalization is that the process of wall enlargement in general is associated with obligatory turnover of the peptidoglycan.     

Rate and time of DNA synthesis of individual Chinese hamster cells.

The duration of DNA synthesis of a diploid cell line of Chinese hamster fibroblasts was determined in a comparative study by the FLM technique, and also by a new technique for measuring the rate of DNA synthesis of individual cells. These methods produced comparable results when applied during exponential growth of the cells. The rate of DNA synthesis was measured by means of quantitative autoradiography following a short-term incubation of the cells with 5 X 10(-6) M FUdR and 10(-5) M 14C-TdR. The choice of the medium for this purpose did not seem to be critical. The autoradiographic silver grains over cells and 14C-standard sources are counted by microphotometry using incident light bright-field. The direct measurements of DNA synthesis rate are 'compartment' statistics which have been converted into 'flux' parameters for comparison with the FLM method and applicability in cell-kinetic calculations. Frequency distributions of the rate of DNA synthesis of individual cells thus obtained may resemble normal distributions quite closely. They result from several factors: differences in the rate of synthesis in different parts of the S-phase, the density distribution of cells within the S-phase, the variation in the time of DNA synthesis among individual cells, and the experimental error. In the case of a pronounced partial synchronization as probably has been present in one experiment performed in the lag phase, an incorrect time of DNA synthesis may result from the rate values. Due to the variation in DNA synthesis rate in different parts of the S-phase it is not possible to determine the duration of DNA synthesis of an individual cell. However, the mean values of DNA synthesis time are reliable. The new method will be preferentially applied for determining the duration of DNA synthesis of human cells in as far as difficulties are encountered with the classical methods. In addition, it may be used to advantage for studying cells which make up low percentages in mixed populations. It finally permits a safer morphological classification of the cells under study than is possible with the classical methods.     

Regulation of RNA synthesis in yeast III

Summary Centrifugal elutriation was used to separate cells in different stages of the cell cycle from a culture of Saccharomyces cerevisiae in balanced exponential growth. The rate of DNA and RNA synthesis was determined using a pulse-long-term label technique that is capable of distinguishing between exponential, linear, and periodic variations in the rate of synthesis through the cell cycle. It was found that while the rate of DNA synthesis varies periodically through the cell cycle, the rate of synthesis of mRNA, rRNA, and tRNA increases exponentially through the cell cycle. The implications of these findings for the control of RNA synthesis are discussed.     

A count-dependent filter for smoothing flow cytometric histograms

An adaptive count-dependent algorithm for smoothing statistically limited histograms has been developed. It considers both the spatial frequency limitations of the measurement system (described by the measurement system point spread function) and the reliability of the measured data (indicated by the effective number of counts influencing each channel of the histogram. Windows for smoothing flow cytometric histograms are derived from an assumed Gaussian-shaped point spread function (PSF) with a constant coefficient of variation. The windows are developed by scaling the variances of the Gaussian functions inversely with the statistical reliability of the data contained in each channel of the measured histogram. The reliability of this data is determined by taking the square root of the number of counts influencing the value tabulated for each channel. Using the algorithm, a smoothed version of the measured histogram may be developed from a linear sum of the products of the individual scaled Gaussian functions and the original measured histogram. Data are presented demonstrating the advantages of count-dependent smoothing over non-count-dependent smoothing using synthesized DNA histograms as a function of sample size.     

RATE AND TIME OF DNA SYNTHESIS OF INDIVIDUAL CHINESE HAMSTER CELLS

The duration of DNA synthesis of a diploid cell line of Chinese hamster fibroblasts was determined in a comparative study by the FLM technique, and also by a new technique for measuring the rate of DNA synthesis of individual cells. These methods produced comparable results when applied during exponential growth of the cells. The rate of DNA synthesis was measured by means of quantitative autoradiography following a short-term incubation of the cells with 5 × 10^-6 M FUdR and 10^-5 M ¹⁴C-TdR. The choice of the medium for this purpose did not seem to be critical. The autoradiographic silver grains over cells and ¹⁴C-standard sources are counted by microphotometry using incident light bright-field. The direct measurements of DNA synthesis rate are ‘compartment’ statistics which have been converted into ‘flux’ parameters for comparison with the FLM method and applicability in cell-kinetic calculations. Frequency distributions of the rate of DNA synthesis of individual cells thus obtained may resemble normal distributions quite closely. They result from several factors: differences in the rate of synthesis in different parts of the S-phase, the density distribution of cells within the S-phase, the variation in the time of DNA synthesis among individual cells, and the experimental error. In the case of a pronounced partial synchronization as probably has been present in one experiment performed in the lag phase, an incorrect time of DNA synthesis may result from the rate values. Due to the variation in DNA synthesis rate in different parts of the S-phase it is not possible to determine the duration of DNA synthesis of an individual cell. However, the mean values of DNA synthesis time are reliable. The new method will be preferentially applied for determining the duration of DNA synthesis of human cells in as far as difficulties are encountered with the classical methods. In addition, it may be used to advantage for studying cells which make up low percentages in mixed populations. It finally permits a safer morphological classification of the cells under study than is possible with the classical methods.     

Cell Growth and Division: III. Conditions for Balanced Exponential Growth in a Mathematical Model

In a previous paper, we proposed a model in which the volume growth rate and probability of division of a cell were assumed to be determined by the cell's age and volume. Some further mathematical implications of the model are here explored. In particular we seek properties of the growth and division functions which are required for the balanced exponential growth of a cell population. Integral equations are derived which relate the distribution of birth volumes in successive generations and in which the existence of balanced exponential growth can be treated as an eigenvalue problem. The special case in which all cells divide at the same age is treated in some detail and conditions are derived for the existence of a balanced exponential solution and for its stability or instability. The special case of growth rate proportional to cell volume is seen to have neutral stability. More generally when the division probability depends on age only and growth rate is proportional to cell volume, there is no possibility of balanced exponential growth. Some comparisons are made with experimental results. It is noted that the model permits the appearance of differentiated cells. A generalization of the model is formulated in which cells may be described by many state variables instead of just age and volume.     

Molar growth yield of Nitrobacter winogradskyi during exponential growth

A numerical analysis of experimental growth curves obtained for Nitrobacter by observing changes in cell numbers, substrate concentration and rate of heat evolution has allowed the calculation of the growth rate constants during the phase of balanced growth. The molar growth yield was smaller during that phase than during the phase preceding it. On the other hand, the rate of heat evolution was larger during exponential growth by a factor of about 1.5 than during the stages up to and including this phase. The two observations being in agreement since, if less efficient synthesis occurs during exponential growth, more free energy must be dissipated as heat.