Estimating the Total Rate of DNA Replication Using Branching Processes

Increasing the knowledge of various cell cycle kinetic parameters, such as the length of the cell cycle and its different phases, is of considerable importance for several purposes including tumor diagnostics and treatment in clinical health care and a deepened understanding of tumor growth mechanisms. Of particular interest as a prognostic factor in different cancer forms is the S phase, during which DNA is replicated. In the present paper, we estimate the DNA replication rate and the S phase length from bromodeoxyuridine-DNA flow cytometry data. The mathematical analysis is based on a branching process model, paired with an assumed gamma distribution for the S phase duration, with which the DNA distribution of S phase cells can be expressed in terms of the DNA replication rate. Flow cytometry data typically contains rather large measurement variations, however, and we employ nonparametric deconvolution to estimate the underlying DNA distribution of S phase cells; an estimate of the DNA replication rate is then provided by this distribution and the mathematical model.     

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.     

Estimating the distribution of the G(2) phase duration from flow cytometric histograms

A mathematical model, based on branching processes, is proposed to interpret BrdUrd DNA FCM-derived data. Our main interest is in determining the distribution of the G(2) phase duration. Two different model classes involving different assumptions on the distribution of the G(2) phase duration are considered. Different assumptions of the G(2) phase duration result in very similar distributions of the S phase duration and the estimated means and standard deviations of the G(2) phase duration are all in the same range.     

A Markov model approach shows a large variation in the length of S phase in MCF-7 breast cancer cells.

BACKGROUND: The potential doubling time of a tumor has been suggested to be a measurement of tumor aggressiveness; therefore, it is of interest to find reliable methods to estimate this time. Because of variability in length of the various cell cycle phases, stochastic modeling of the cell cycle might be a suitable approach. METHODS: The relative movement curve and the DNA synthesis time were estimated by using local polynomial regression methods. Further, the rate of nucleotide incorporation was estimated by using a Markov pure birth process with one absorbing state to model the progression of the DNA distribution through S phase. RESULTS: An estimate of the DNA synthesis time, with confidence intervals, was obtained from the relative movement curve. The Markov approach provided an estimate of the distribution of the time to complete S phase given the initial distribution. Using the Markov approach we also made an estimate of the mean number of active replicons during S phase. CONCLUSIONS: A Markov pure birth process has shown to be useful to model the progression of cells through S phase and to increase knowledge about the variability in the length of S phase and a large variation is shown.     

A population balance model describing the cell cycle dynamics of myeloma cell cultivation

A multi-staged population balance model is proposed to describe the cell cycle dynamics of myeloma cell cultivation. In this model, the cell cycle is divided into three stages, i.e., G1, S, and G2M phases. Both DNA content and cell volume are used to differentiate each cell from other cells of the population. The probabilities of transition from G1 to S and division of G2M are assumed to be dependent on cell volume, and transition probability from S to G2M is determined by DNA content. The model can be used to simulate the dynamics of DNA content and cell volume distributions, phase fractions, and substrate and byproduct concentrations, as well as cell densities. Measurements from myeloma cell cultivations, especially the FACS data with respect to DNA distribution and cell fractions in different stages, are employed for model validation.     

Influence of spectral wavelength and N:P ratio on the cell cycle of Synechococcus strain CSIRNIO1 isolated from the central Eastern Arabian Sea

The novel phycoerythrin-containing Synechococcus strain CSIRNIO1 belonging to phylogenetic clade II was isolated from the coastal Arabian Sea. Chromophore characteristics of this isolate revealed the presence of phycoerythrin I (PEI), which allows it to utilize green light efficiently. The DNA distribution data indicate a bimodal slow growth model synchronized with the light/dark cycle. The duration of the cell cycle was regulated by spectral wavelength and nutrient concentration. Nitrate and phosphate enrichment shortened G1 phase duration when cells were exposed to equal doses of photosynthetically usable radiation (PUR) of different spectral wavelengths. G2 phase duration was influenced by spectral quality and phosphate concentration. S phase duration was not affected by the spectral wavelength. However, a shorter doubling time corresponding to shortened G1 and S phases was observed under nitrate enrichment. Phosphate enrichment resulted in shortening of all three phases (G1, S and G2). More efficient utilization of green and red light than blue light regulated the duration of the cell cycle as well as the doubling time, suggesting spectral selectivity in this strain. The effects of spectral wavelengths under varying nutrient concentrations will determine the proliferation of Synechococcus and its adaptation to different environmental conditions.     

Influence of the existence of a resting state on the probability of cell division in culture

A cell cycle model developed by Smith and Martin is generalized to allow for the possibility that the duration of the B phase is not fixed. The B phase is the equivalent of the traditional S, G₂, and M phases of the cell cycle. The duration of the B phase is represented by a Gaussian probability distribution; the duration of the resting or A state which replaces the traditional G₁ phase is represented by a decaying exponential distribution. A doubling time distribution, termed the CEG distribution, is obtained by convolution of the A state and B phase distributions. Like the reciprocal normal, rate normal, and log normal distributions, it is a rounded unimodal peak that is skewed to the right. None of the three former distributions is associated with a cell cycle model that includes a resting state. However the CEG distribution, which is so associated, bears little resemblance to the delayed exponential distribution which results when the duration of the B phase is fixed and the duration of the A state is random. Consequently, it would be difficult to use the doubling time distribution to determine whether or not a resting state exists in a particular cell population.     

The cytoplasmic origin of variability in the timing of S phase in mammalian cells.

The time at which S phase begins in mammalian cells is highly variable with respect to cell age. Evidence is presented that this variability does not arise because the initiation of DNA synthesis depends on the stochastic interaction of an initiator substance with a rare initiation site. Instead, the signal responsible for starting S phase must appear at random in the cytoplasm and may be transient.     

Mathematical modelling of whole chromosome replication

All chromosomes must be completely replicated prior to cell division, a requirement that demands the activation of a sufficient number of appropriately distributed DNA replication origins. Here we investigate how the activity of multiple origins on each chromosome is coordinated to ensure successful replication. We present a stochastic model for whole chromosome replication where the dynamics are based upon the parameters of individual origins. Using this model we demonstrate that mean replication time at any given chromosome position is determined collectively by the parameters of all origins. Combining parameter estimation with extensive simulations we show that there is a range of model parameters consistent with mean replication data, emphasising the need for caution in interpreting such data. In contrast, the replicated-fraction at time points through S phase contains more information than mean replication time data and allowed us to use our model to uniquely estimate many origin parameters. These estimated parameters enable us to make a number of predictions that showed agreement with independent experimental data, confirming that our model has predictive power. In summary, we demonstrate that a stochastic model can recapitulate experimental observations, including those that might be interpreted as deterministic such as ordered origin activation times.     

Analysis of a model of cell cycle in eukaryotes

A dynamic model of the cell cycle for eukaryotic cells, which takes into account the rates of ribosome and protein synthesis and the discontinuous events of DNA replication and cell division, is analyzed. It is shown that, by changing the values of the parameters, three different cell cycle regimens are possible, which are similar to cell cycle patterns experimentally observed and which show the action of different control mechanisms. The model allows the determination of the macromolecular levels as a function of the cycle time. Taking into consideration the age distribution function of the cells in an ideal exponentially growing population, mathematical relations are calculated that link the levels of macromolecular components (protein, ribosomes and DNA) to the temporal parameters of the cell cycle, such as the relative duration of the S phase. It is also shown that the relative length of all cell cycle phases may be determined if the labelling index and the relative DNA content of the cell population are known. All these relations suggest new and convenient procedures to determine cell cycle parameters.     

Double label estimation of the mean duration of the S-phase

The theoretical values of the double label index and the estimator of the mean duration of the DNA-synthesis phase are obtained for the situation where a steady-state renewal cell population is subjected to two instantaneous labels. The stochastic model used to describe this situation allows for a correlation structure among the durations of the phases of the cell cycle. An explicit form for the double label index is presented for the case where the durations of the phases of the cell cycle are jointly distributed in accordance with a multivariate mixed gamma distribution. The effect of the length sampling bias and the time between label points on the theoretical value of the estimator is demonstrated.     

Correlation between cell cycle duration and RNA content.

The metachromatic fluorochrome acridine orange was used to differentially stain DNA and RNA in Chinese hamster ovary (CHO) cells and in mitogen-stimulated human lymphocytes during their progression through the cell cycle. Green and red fluorescence of individual cells, representing cellular DNA and RNA, respectively, was measured by flow cytometry. CHO cells were synchronized by selective detachment at mitosis. Their rate of progression through G1 and subsequently through S phase correlated with the content of stainable RNA. The mean duration of the G1 phase was 5.2 hours for cells with high RNA content (highest 25 percentile population) and 8.1 hours for cells with low RNA (lowest 25 percentile). The duration of S phase was 5.9 and 7.5 hours for high- and low-RNA, 25 percentile subpopulations, respectively. Lymphocytes synchronized at the G1/S boundary by hydroxyurea or 5-fluorodeoxyuridine showed extremely high intercellular variation with respect to content of stainable RNA. After release from the block they traversed S phase at rates linearly proportional to the content of stainable RNA. The duration of S phase was five hours for cells with high RNA-, six to nine hours for cells with moderate RNA- and up to 27 hours for cells with minimal RNA-content. The data suggest that the rate of progression through the cell cycle of individual cells within a population may be correlated with the number of ribosomes per cell.     

Effects of inhibition of RNA or protein synthesis on CHO cell cycle progression

Chinese hamster ovary (CHO) cells, synchronized by selective detachment at mitosis, were treated with various concentrations of actinomycin D (AMD) or cycloheximide (CHX) either immediately, or 1, 2, or 3 hr after mitosis. Since the minimum duration of G₁ phase in these cultures was 3.4 hr, the addition of RNA or protein synthesis inhibitors took place at the beginning, first third, second third, or end (G₁–S boundary) of G₁ phase. The kinetics of exit from G₁ phase, the rate and extent of traverse of S phase, and the reaccumulation of RNA were estimated under each set of growth conditions by flow cytometry of acridine orange-stained cells. A mathematical model was constructed to describe the trajectories of the cell populations with respect to their increase in RNA and DNA content in the absence or presence of the inhibitor. The chronologic synchrony imposed on the CHO cell population began to decay within 3 hr, resulting in stochastic entrance of cells into S phase in the absence of inhibitor. Addition of AMD or CHX at 0, 1, 2, or 3 hr after mitosis, regardless of the inhibitor concentration, did not provide evidence of a critical restriction point in G₁ beyond which cells were committed to enter S phase and were no longer sensitive to moderate suppression of RNA or protein synthesis. The observed kinetics of cell entrance into and traverse of S phase were consistent with an inherently heterogenous response to serum stimulation occurring at or just after cell division.     

Influence of chromatin structure on induction of double-strand breaks in mammalian cells irradiated with DNA-incorporated 125I

In this study the induction of double-strand breaks (DSBs) was investigated in Chinese hamster V79-379A cells irradiated with the Auger-electron emitter (125)I incorporated into DNA. The role of chromatin organization was studied by pulse-labeling synchronized cells with (125)IdU before decay accumulation in early or late S phase. Pulsed-field gel electrophoresis and fragment-size analysis were used to quantify the distribution of DNA fragments in irradiated intact cells and naked DNA as well as in DNA from asynchronously labeled cultures in a different scavenging environment. The results show that in intact cells, after accumulation of decays at -70 degrees C in the presence of 10% DMSO, almost four times more DSBs were induced in late S phase compared with early S phase and the fragment distribution was clearly non-random with an excess of fragments <0.2 Mbp. The DSB yield was 0.6 DSB/cell and decay for cells irradiated in early S phase and 2.3 DSBs/cell and decay for cells irradiated in late S phase. When similar experiments were performed on naked genomic DNA or intact cells irradiated with gamma rays, the difference in yield was not as prominent. These data imply a role of chromatin organization in the induction of DSBs by DNA-incorporated (125)I. In summary, the results presented here suggest that the yield of DSBs as well as the fragment distribution induced by (125)IdU decay may vary significantly depending on the chromatin organization during S phase and the labeling procedure used.     

Stochastic mechanism of cellular aging--abrupt telomere shortening as a model for stochastic nature of cellular aging

A strong stochastic component has been described for the appearance of senescent cells in cultures that have not completed their in vitro lifespan. The proliferative potential of individual clones show a bimodal distribution. Additionally, two cells arising from a single mitotic event can exhibit large differences in their doubling capacities. In this report we present a model and a computer simulation of the model that explains the observed stochastic phenomena. The model is based on both gradual and abrupt telomere shortening.Gradual telomere shortening (GTS) occurs during each cell division as a consequence of the inability of DNA polymerase to replicate the very ends of chromosomal DNA. It is responsible for the gradual decline in proliferative potential of a cell culture, but does not explain the stochastic aspects of cellular aging. Abrupt telomere shortening (ATS) occurs either through DNA recombination or nuclease digestion at the subtelomeric/telomeric border region of the chromosome. Recombination involves the invasion of a telomere single-strand three-prime protruding end at this border in the telomere of the same chromosome or in another subtelomeric/telomeric region. Shortening of one or more telomeres in the cell causes a sudden onset of cell senescence, referred to as sudden senescence syndrome (SSS). This is manifested as a stochastic and abrupt transition of cells from the larger to the smaller proliferative potential pool and can cause cell cycle arrest within one cell division. The computer simulation matches well with experimental data supporting the prediction that abrupt telomere shortening underlies the stochastic onset of cell senescence. Sudden senescence syndrome appears to be the most important mechanism in the control of the extent of proliferation of a cell culture because it prevents virtually every cell in the culture from reaching its maximum doubling capacity, that would otherwise be allowed by telomere shortening via the end-replication mechanism alone.     

The probability of G1 cells to enter into S increases with their size while S length decreases with cell enlargement in Allium cepa

The distribution of cell surface area projection (cell size) has been measured at birth and at initiation of DNA synthesis in steady-state populations of Allium cepa root meristems. The conditional probability, P(I/G1), that initiation occurs given that the event of being in G1 also occurs has been estimated from these data. P(I/G1) was found to increase when cells became larger. The distribution of G1 duration has been constructed from indicated cell size distributions. The absolute frequencies of G1 times showed a maximum in the zone of cells with short G1 periods; about 14% of cells appear to enter into S with G1 congruent to 1 h. These results suggest that the increase of P(I/G1) was due to cell enlargement and not to cell aging. By comparing the cell size distribution at initiation of S and at the end of this period, a drastic reduction of cell size variability during DNA replication was observed and both curves were seen as rather similar in shape although they obviously had different modal points. These observations support that there is a negative correlation between the initiation size and the duration of genome duplication, and that cells which initiate DNA synthesis with the same size have a similar replication time. From this hypothesis, a plot of S duration versus cell size at initiation of this period was constructed by comparing the distributions of cell size at start and end of replication; this plot was also consistent with the existence of a negative correlation between cell initiation size and S length.     

In silico simulation of the effect of hypoxia on MCF-7 cell cycle kinetics under fractionated radiotherapy

The treatment outcome of a given fractionated radiotherapy scheme is affected by oxygen tension and cell cycle kinetics of the tumor population. Numerous experimental studies have supported the variability of radiosensitivity with cell cycle phase. Oxygen modulates the radiosensitivity through hypoxia-inducible factor (HIF) stabilization and oxygen fixation hypothesis (OFH) mechanism. In this study, an existing mathematical model describing cell cycle kinetics was modified to include the oxygen-dependent G1/S transition rate and radiation inactivation rate. The radiation inactivation rate used was derived from the linear-quadratic (LQ) model with dependence on oxygen enhancement ratio (OER), while the oxygen-dependent correction for the G1/S phase transition was obtained from numerically solving the ODE system of cyclin D-HIF dynamics at different oxygen tensions. The corresponding cell cycle phase fractions of aerated MCF-7 tumor population, and the resulting growth curve obtained from numerically solving the developed mathematical model were found to be comparable to experimental data. Two breast radiotherapy fractionation schemes were investigated using the mathematical model. Results show that hypoxia causes the tumor to be more predominated by the tumor subpopulation in the G1 phase and decrease the fractional contribution of the more radioresistant tumor cells in the S phase. However, the advantage provided by hypoxia in terms of cell cycle phase distribution is largely offset by the radioresistance developed through OFH. The delayed proliferation caused by severe hypoxia slightly improves the radiotherapy efficacy compared to that with mild hypoxia for a high overall treatment duration as demonstrated in the 40-Gy fractionation scheme.     

Universal Temporal Profile of Replication Origin Activation in Eukaryotes

Although replication proteins are conserved among eukaryotes, the sequence requirements for replication initiation differ between species. In all species, however, replication origins fire asynchronously throughout S phase. The temporal program of origin firing is reproducible in cell populations but largely probabilistic at the single-cell level. The mechanisms and the significance of this program are unclear. Replication timing has been correlated with gene activity in metazoans but not in yeast. One potential role for a temporal regulation of origin firing is to minimize fluctuations in replication end time and avoid persistence of unreplicated DNA in mitosis. Here, we have extracted the population-averaged temporal profiles of replication initiation rates for S. cerevisiae, S. pombe, D. melanogaster, X. laevis and H. sapiens from genome-wide replication timing and DNA combing data. All the profiles have a strikingly similar shape, increasing during the first half of S phase then decreasing before its end. A previously proposed minimal model of stochastic initiation modulated by accumulation of a recyclable, limiting replication-fork factor and fork-promoted initiation of new origins, quantitatively described the observed profiles without requiring new implementations.The selective pressure for timely completion of genome replication and optimal usage of replication proteins that must be imported into the cell nucleus can explain the generic shape of the profiles. We have identified a universal behavior of eukaryotic replication initiation that transcends the mechanisms of origin specification. The population-averaged efficiency of replication origin usage changes during S phase in a strikingly similar manner in a highly diverse set of eukaryotes. The quantitative model previously proposed for origin activation in X. laevis can be generalized to explain this evolutionary conservation.