Estimation of parameters in population models for Schizosaccharomyces pombe from chemostat data

The integro-differential growth model of Eakman, Fredriekson, and Tsuehiya has been employed to fit cell size distribution data for Schizosaccharomyces pombe grown in a chemostat under severe product inhibition by ethanol. The distributions were obtained with a Coulter aperture and an electronic system patterned after that of Harvey and Marr. Four parameters—mean cell division size, cell division size standard deviation, daughter cell size standard deviation, and a growth rate coefficient—were calculated for models where the cell growth rate was inversely proportional to size, constant, and proportional to size. A fourth model, one where sigmoidal growth behavior was simulated by two linear growth segments, was also investigated. Linear and sigmoidal models fit the distribution data best. While the mean cell division size remained relatively constant at all growth rates, standard deviation of division size distribution increased with increasing holding times. Standard deviation of the daughter size distribution remained small at all dilution rates. Unlike previous findings with other organisms, the average cell size of Schizosaccharomyces pobme increased at low growth rates.     

The geometry of elongating plant cells

Shape and size of elongating cells were examined in three plant tissues: the adaxial epidermis of the petiole ofZebrina pendula L., the abaxial epidermis ofAnacharis densa L. leaves and the abaxial epidermis of the scale leaf ofAllium cepa L. Based on a few simple assumptions, the expected probability distribution frequencies (pdf) for cell length and number of adjacent walls were calculated. Actual data of cell lengths closely approximated those expected with the pdfs being asymmetrical since there are more younger, shorter cells than older, longer cells. Data for number of lateral walls of real cells were similar to that expected and these walls increase in compensating mechanism exists to maintain a constant range of cell lengths through many cell generations. It is expressed by longer than average new daughter cells dividing relatively soon while shorter than average new daughter cells divide after a relatively long cycle.     

Size control models of Saccharomyces cerevisiae cell proliferation.

By using time-lapse photomicroscopy, the individual cycle times and sizes at bud emergence were measured for a population of saccharomyces cerevisiae cells growing exponentially under balanced growth conditions in a specially constructed filming slide. There was extensive variability in both parameters for daughter and parent cells. The data on 162 pairs of siblings were analyzed for agreement with the predictions of the transition probability hypothesis and the critical-size hypothesis of yeast cell proliferation and also with a model incorporating both of these hypotheses in tandem. None of the models accounted for all of the experimental data, but two models did give good agreement to all of the data. The wobbly tandem model proposes that cells need to attain a critical size, which is very variable, enabling them to enter a start state from which they exit with first order kinetics. The sloppy size control model suggests that cells have an increasing probability per unit time of traversing start as they increase in size, reaching a high plateau value which is less than one. Both models predict that the kinetics of entry into the cell division sequence will strongly depend on variability in birth size and thus will be quite different for daughters and parents of the asymmetrically dividing yeast cells. Mechanisms underlying these models are discussed.     

Effects of Chromosome Underreplication on Cell Division in Escherichia coli

The key processes of the bacterial cell cycle are controlled and coordinated to match cellular mass growth. We have studied the coordination between replication and cell division by using a temperature-controlled Escherichia coli intR1 strain. In this strain, the initiation time for chromosome replication can be displaced to later (underreplication) or earlier (overreplication) times in the cell cycle. We used underreplication conditions to study the response of cell division to a delayed initiation of replication. The bacteria were grown exponentially at 39°C (normal DNA/mass ratio) and shifted to 38 and 37°C. In the last two cases, new, stable, lower DNA/mass ratios were obtained. The rate of replication elongation was not affected under these conditions. At increasing degrees of underreplication, increasing proportions of the cells became elongated. Cell division took place in the middle in cells of normal size, whereas the longer cells divided at twice that size to produce one daughter cell of normal size and one three times as big. The elongated cells often produced one daughter cell lacking a chromosome; this was always the smallest daughter cells, and it was the size of a normal newborn cell. These results favor a model in which cell division takes place at only distinct cell sizes. Furthermore, the elongated cells had a lower probability of dividing than the cells of normal size, and they often contained more than two nucleoids. This suggests that for cell division to occur, not only must replication and nucleoid partitioning be completed, but also the DNA/mass ratio must be above a certain threshold value. Our data support the ideas that cell division has its own control system and that there is a checkpoint at which cell division may be abolished if previous key cell cycle processes have not run to completion.     

Cell cycle dynamics inferred from the static properties of cells in balanced growth

The duration of a morphological phase of the cell cycle is reflected in the steady state distribution of the sizes of cells in that phase. Relationships presented here provide a method for estimating the timing and variability of any cell cycle phase. It is shown that the mean size of cells initiating and finishing any phase can be estimated from (1) the frequency of cells exhibiting the distinguishing morphological or autoradiographic features of the phase; (2) the mean size of cells in the phase; and (3) their coefficient of variation. The calculations are based on a submodel of the Koch-Schaechter Growth Controlled Model which assumes that (i) the distribution of division sizes is Gaussian; (ii) there is no correlation in division sizes between successive generations; and (iii) every cell division gives rise to two daughter cells of equal size. The calculations should be useful for a wider range of models, however, because the extrapolation factors are not sensitive to the chosen model. Criteria are proposed to allow the user to check the method's applicability for any experimental case. The method also provides a more efficient test of the dependence of growth on cell size than does the Collins-Richmond method. This is because the method uses the mean and coefficient of variation of the size of the total population, in conjunction with those of the cells in a final phase of the cell cycle, to test potential growth laws. For Escherichia coli populations studied by electron microscopy, an exponential growth model provided much better agreement than did a linear growth model. The computer simulations were used to generate rules for three types of cell phases: those that end at cell division, those that start at cell division, and those totally contained within a single cell cycle. For the last type, additional criteria are proposed to establish if the phase is well enough contained for the formulae and graphs to be used. The most useful rule emerging from these computer studies is that the fraction of the cell cycle time occupied by a phase is the product of the frequency of the phase and the ratio of the mean size of cells in that phase to the mean size of all cells in the population. A further advantage of the techniques presented here is that they use the 'extant' distributions that were actually measured, and not hypothesized distributions nor the special distributions needed for Collins-Richmond method that can only be calculated from the observed distributions of dividing or newborn cells on the basis of an assumed growth law.     

Analysis of the cell cycle duration of cells of permanent line L-929

The direct measurement of the cell cycle duration in L-929 cells was performed using time-lapse photography. The cell cycle duration was 15.77 +/- 0.08 h with a standard deviation of 1.54 +/- 0.06 h. The experimental value fit to a normal distribution with a correlation coefficient 0.999. High homogeneity of this parameter and a wide range of variability of the karyotype (58-66 chromosomes) indicate that there is no correlation between these characteristics of L-929 cells. It is also shown that the difference between cell cycle durations of daughter cells tent to zero and fits by an exponent.     

Distribution of mutants in aggregates of cultured cells

The analysis of the distribution of mutants in an exponentially growing culture of cells that are aggregated into clumps of homogeneous size is described, given the mutation rate and a random process by which clumps divide to produce progeny. The mean and standard deviation of the proportion of clumps with a given number of mutant cells at a particular time are calculated. Since the standard deviation tends to be much smaller than the mean, the following conclusions can be drawn. Aggregation lowers the number of mutant-containing clumps in cultures grown to a standard number of cells, but raises the number of mutant-containing clumps in cultures grown to a standard number of clumps. In the absence of mutation, or at low mutation rates, clumps tend to become pure types (normal or mutant). The probability of finding pure, nonmutant-containing clumps, however, is approximately the initial fraction of nonmutant cells (given realistic forward and back mutation rates). Also, in terms of the given process, it is possible to compute the probability that all the cells in an aggregate descend from a single, common parent cell within a given number of generations, and thus to calculate the probability that all the cells in a clone grown from an aggregate descend from a single cell within a known number of generations.     

Volume growth of daughter and parent cells during the cell cycle of Saccharomyces cerevisiae a/alpha as determined by image cytometry.

The pattern of volume growth of Saccharomyces cerevisiae a/alpha was determined by image cytometry for daughter cells and consecutive cycles of parent cells. An image analysis program was specially developed to measure separately the volume of bud and mother cell parts and to quantify the number of bud scars on each parent cell. All volumetric data and cell attributes (budding state, number of scars) were stored in such a way that separate volume distributions of cells or cell parts with any combination of properties--for instance, buds present on mothers with two scars or cells without scars (i.e., daughter cells) and without buds--could be obtained. By a new method called intersection analysis, the average volumes of daughter and parent cells at birth and at division could be determined for a steady-state population. These volumes compared well with those directly measured from cells synchronized by centrifugal elutriation. During synchronous growth of daughter cells, the pattern of volume increase appeared to be largely exponential. However, after bud emergence, larger volumes than those predicted by a continuous exponential increase were obtained, which confirms the reported decrease in buoyant density. The cycle times calculated from the steady-state population by applying the age distribution equation deviated from those directly obtained from the synchronized culture, probably because of inadequate scoring of bud scars. Therefore, for the construction of a volume-time diagram, we used volume measurements obtained from the steady-state population and cycle times obtained from the synchronized population. The diagram shows that after bud emergence, mother cell parts continue to grow at a smaller rate, increasing about 10% in volume during the budding period. Second-generation daughter cells, ie., cells born from parents left with two scars, were significantly smaller than first-generation daughter cells. Second- and third-generation parent cells showed a decreased volume growth rate and a shorter budding period than that of daughter cells.     

Age Distribution of Human Diploid Fibroblasts: A Stochastic Model for In Vitro Aging

Variation in the lifespan of mass cultures and clones of human diploid fibroblasts can be explained on the basis of variation in the length of the mitotic cycle. This variation is of biological significance; the intrinsic standard deviation of culture lifespan is equal to about 10% of the mean. We constructed a two-parameter stochastic model based on the following assumptions: the time between successive divisions of a given cell is of random duration; cells divide or lose the ability to divide independently of one another; the probability that a cell can undergo further division is constant up to some maximum number of divisions and zero thereafter. We determined numerically the proportion of nondividing cells and the distribution of cell generations. Samples taken by Monte Carlo means from a hypothetical in vitro population were compared with clonal survival data obtained experimentally. The fit between experimental and theoretical findings was within the range of sampling variation. If we accept our model as being applicable to human diploid cell culture, we can draw the following conclusions: the proportion of dividing cells is an inadequate index of a population's age; even in populations in which almost all cells are still capable of division, a majority of the cells have less than eight generations remaining to them. At each subcultivation the ultimate fate of a culture is determined by the disposition of a relatively small number of “young” cells.     

Kinetic Cell-Cycle Analysis of a Cultured Mammalian Cell Population

The parameters of the cell cycle are analyzed in terms of the stochastic theory of cell proliferation for a murine mastocytoma line. The cells were grown in suspension culture under steady-state conditions in a chemostat. Initial estimates of the parameters from synchronous growth indicate that agreement of the data with the model is obtained only if the model is modified to include an initial proliferating fraction of less than 100%, and a cell loss continuing throughout the course of the experiment. The analysis verifies that the modified theory adequately describes the data, and that similar parameters are obtained from both desynchronization and percent labeled mitosis experiments. The average cycle time from 10 desynchronization experiments was 8.24 ± 0.52 h with a cellular standard deviation of 1.28 ± 0.18. The combined parameter obtained by dividing the cellular standard deviation by the cycle time is shown to be a useful measure of biological variability well defined over many different experiments. The rate constant for cell loss is about 0.009 which gives an 8% cell loss per cycle. The cell loss is sufficient to account for the apparent deficit in initially proliferating cells. The initial distribution of the synchronous cells is qualitatively examined and is found to be peaked late in G₁ or early in S.     

Analysis of cell proliferation in Drosophila wing imaginal discs using mosaic clones

Experimental data on spatial and temporal distributions of mosaic clones in Drosophila wing imaginal disc were analyzed. Long-lived proliferation centers (PR1, PR2, and PR3) and areas with decreased proliferation activity were found in the notum region of the disc. Simulation of the growth kinetics of mosaic patches demonstrated that the cell cycle in proliferation centers PR2 and PR3 was shorter than the average cycle in the disc and in the center PR1. A nonrandom clustering of rapidly dividing cells was observed in the PR2, but not in the other cases. The reason why the cell-cycle duration and the clustering of dividing cells may not coincide is discussed in terms of the recruitment of nondividing cells into the cell cycle. The simulation of the time course of the first and second moments of the size distribution of mosaic clones allowed the variance of cell-cycle progression rates to be determined and demonstrated that a model with a continuous cell-cycle rates gave a better fit to the data than the transition probability model of Smith and Martin.     

Probability Density Function of the Red Cell Membrane Permeability Coefficient

The distribution of a random variable is determined by the probability density functions (PDF) of all other random variables with which the variable in question is jointly distributed. If the PDF of the random variable of interest is normal, or skewed normal, then the distributions with which it is jointly distributed determine its mean and standard deviation. In the case described here (where hemolysis time of the red blood cell is a function of the permeability coefficient and geometric variables of the cell) the mean and standard deviation of the permeability coefficient and the known distributions of the geometric variables on which the hemolysis time depends determine a predicted distribution of hemolysis time. An observed distribution of the hemolysis time is obtained spectrophotometrically. By choosing the mean and standard deviation of the permeability coefficient so that the predicted PDF of the hemolysis time matches the observed PDF best by least-squares criterion, the complete distribution of the permeability coefficient is determined.     

Bromodeoxyuridine labeling and flow cytometric identification of replicating Saccharomyces cerevisiae cells: lengths of cell cycle phases and population variability at specific cell cycle positions.

An immunofluorescent staining procedure has been developed to identify, with flow cytometry, replicating cells of Saccharomyces cerevisiae after incorporation of bromodeoxyuridine (BrdUrd) into the DNA. Incorporation of BrdUrd is made possible by using yeast strains with a cloned thymidine kinase gene from the herpes simplex virus. An exposure time of 4 min to BrdUrd results in detectable labeling of the DNA. The BrdUrd/DNA double staining procedure has been optimized and the flow cytometry measurements yield histograms comparable to data typically obtained for mammalian cells. On the basis of the accurate assessment of cell fractions in individual cell cycle phases of the asynchronously growing cell population, the average duration of the cell cycle phases has been evaluated. For a population doubling time of 100 min it was found that cells spend in average 41 min in the replicating phase and 24 min in the G2+M cell cycle period. Assuming that mother cells immediately reenter the S phase after cell division, daughter cells spend 65 min in the G1 cell cycle phase. Together with the single cell fluorescence parameters, the forward-angle light scattering intensity (FALS) has been determined as an indicator of cell size. Comparing different temporal positions within the cell cycle, the determined FALS distributions show the lowest variability at the beginning of the S phase. The developed procedure in combination with multiparameter flow cytometry should be useful for studying the kinetics and regulation of the budding yeast cell cycle.     

Stochastic branching model for hemopoietic progenitor cell differentiation

We present algebraic expressions describing the predictions of a stochastic branching model for differentiation of hemopoietic progenitor cells. The model assumes that there is a fixed probability, p (0 less than or equal to p less than or equal to 1), that commitment to a differentiative event occurs per progenitor cell division for each daughter cell. The model describes properties of in vitro hemopoietic cell differentiation including the population structure at the time the first progenitor cell becomes committed, the number of committed progenitor cells engendered by a single progenitor cell, and the probability of eventual commitment of all daughter cells derived from a single progenitor or stem cell. Application of the model to experimental data obtained from erythroid cultures suggests that the observed data can be explained by the stochastic branching model alone without making the deterministic assumption that there is a differentiative hierarchy in the lineage of the progenitors of erythropoiesis (BFU-E). The qualitative and quantitative aspects of the proposed stochastic model are discussed in conjunction with other analogous stochastic branching models.     

Average cell size is a factor reflecting the interaction of CHO cells during their proliferation

Changes in cell area during CHO cell cultivation have been studied with time-lapse microscopy. Capture was started the day after cell plating. It was found that the size of daughter cells after mitosis remained less than the size of the mother cell for a long time (up to 6 h). Nevertheless, the average cell area of the whole population was constant during the observation period (up to 18 h). We assume that this phenomenon is a result of interaction between dividing and nondividing cells. The experimental data we obtained confirm this conclusion.     

Persistence of cell cycle times over many generations as determined by heritability of colony sizes of ras oncogene-transformed and non-transformed cells

Abstract. The persistence of cell lifetimes during about 10 successive cell generations was investigated by comparing the number of cells in primary colonies and in secondary colonies derived from primary colonies. Primary colonies were grown from single cells for 3 or 4 days (a time equivalent to an average of five cell generations) and the number of cells in each primary colony determined. Cells in each primary colony were dispersed to initiate secondary colonies, grown for the same time, and the number of cells in secondary colonies determined. Several criteria were used to compare primary and related secondary colonies, the most informative was found to be regression and correlation coefficients between number of cells in primary colonies and mean numbers of cells in related secondary colonies. For two non-transformed mouse fibroblast cell lines, NIH 3T3 and BALB 3T3, the regression and correlation coefficients of cell number in primary and secondary colonies were positive. This suggests inheritance of cell lifetimes over many cell generations. After the addition of an activated ras oncogene (human cellular Harvey ras , or viral Kirsten ras ) some regression and correlation coefficients changed in magnitude but all remained positive. Comparison of experimental data and the results of computer simulations suggest that several models of inheritance of cell lifetimes are not adequate to explain the results, including a model of independence between lifetimes of mother and daughter cells and the common model that describes daughter cells as inheriting the lifetime of their mother with deviation. Simulations do suggest that cell lifetimes are inherited within clones as deviation from the lifetime of the initial cell, and that the ras oncogene does not destroy persistence within clones but does increase heterogeneity of cell lifetimes.     

Phase of the circadian clock is accurately transferred from mother to daughter cells in the dinoflagellate Gonyaulax polyedra.

In the first cycle following transfer from a 12 h light-12 h dark cycle (LD12:12) to constant darkness (DD), the standard deviation in circadian phase among individual clocks in populations of Gonyaulax polyedra is approximately 60 min. When a culture is transferred to constant light conditions (LL) from an LD 12:12 cycle, the standard deviation increases in the first 2-3 d, but then remains unchanged, suggesting a lack of observable desynchronization in LL after the transient period. The synchrony in a cell population is preserved even after several cell divisions. The results indicate that variations in period among cells are small, that the period of an individual clock does not fluctuate randomly from day to day, and that the circadian phase of a mother cell is faithfully passed to the clocks of the daughter cells.     

Cytomorphometry. A methodologic study of preparation techniques, selection methods and sample sizes

The influence of methodologic aspects on cytomorphometric features was studied using preparations of hepatoma and/or mastocytoma cells. First, two preparation techniques (smear and oese) were compared. Second, four methods of selecting cells for cytomorphometric analysis (two conventional and two stratified methods) were tested for reproducibility. Third, heterogeneous cell populations were used to estimate the required sample size using the running coefficient of variation (CV), and the results were compared with expected (theoretical) values of the required sample size calculated using the standard error of the mean. The results showed significantly lower CVs for the smear preparation technique. The stratified methods appeared to be superior to the conventional methods for selecting cells for measurement. The experimentally assessed sample sizes were considerably lower than the corresponding theoretical calculations. These findings suggest that morphometric assessments in cytologic smears should utilize a stratified cell selection method. While experimentally assessed sample sizes are relatively small and therefore better routinely applicable, they may yield less reliable results in some cases. The need to test a sample for its reproducibility as well as its discriminatory power is emphasized.     

Clustering versus random segregation of plasmids lacking a partitioning function: a plasmid paradox?

Plasmids lacking a functional partition system are randomly distributed to the daughter cells; plasmid-free daughter cells are formed with a frequency of (1/2)2n per cell and cell generation where 2n is the (average) copy number at cell division. Hence, the unit of segregation is one plasmid copy. However, plasmids form clusters in the cells. A putative solution to this potential paradox is presented: one plasmid copy at a time is recruited from the plasmid clusters to the replication factories that are located in the cell centres. Hence, replication offers the means of declustering that is necessary in a growing host population. The daughter copies diffuse freely and each copy may with equal probability end up in either of the two cell halves. In this way, the random segregation of the plasmids is coupled to replication and occurs continuously during the cell cycle, and is not linked to cell division. The unit of segregation is the plasmid copy and not the plasmid clusters. In contrast, the two daughters of a Par+ plasmid are directed in opposite directions by the plasmid-encoded partition system, thereby assuring that each daughter cell receives the plasmid.     

A bimolecular mechanism for the cell size control of the cell cycle

A molecular model for the control of cell size has been developed. It is based on two molecules, one (I) acts as an inhibitor of the entrance into S phase, and it is synthetised just after cell separation in a fixed amount per nucleus. The other (A) is an activator of the S phase, and it is synthetised at a ratio proportional to the overall protein accumulation. The activator reacts stoichiometrically with (I), and after all the (I) molecules have been titrated, (A) begins to accumulate. When it reaches a threshold value, it triggers the onset of DNA replication. This model was tested by simulation and when applied to the case of unequal division explains a number of features of an exponentially growing yeast cell population: (a) the lengths of TP (cycle time of parent cells) and TD (cycle time of daughter cells) verify the condition exp(- KTP ) + exp(- KTD ) = 1; (b) the changes of the average cell size of populations at different growth rates; (c) the frequency of parents and daughters at various growth rates; (d) the increase of cell size at bud initiation for cells of increasing genealogical age; (e) the existence of a TP - TB period (difference between the cycle time of parents and the length of budded phase) that depends linearly upon the doubling time of the population.