Model definition

The eukaryotic cell cycle is defined as an orderly sequence of three phases distinguished by cellular DNA content, termed and A dividing cell is supposed to proceed, under this minimalist view, from one phase to another in a fixed order, until reaching the end of phase. Here it completes cytokinesis generating two genetically identical daughter cells that are by definition in phase (Fig. 1 A). We assume that the completion time of any phase (i.e. the time lapse between the entry to and exit from that given phase) is a random variable which is distributed according to a delayed (or shifted) exponential density function (Fig. 1 B),

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Figure 1. Stochastic cell cycle model.

A: Scheme of the proposed cell cycle model with three phases and The dashed border between the and the phase indicates that the and phase are pooled into a single phase. The random time a cell needs to complete the processes associated to each of the phases, follows a delayed exponential distribution with specific parameters and for each phase. B: Delayed-exponential completion time distribution density with parameters and C: Best fit of the complementary cumulative distribution to the fraction of undivided cells after birth obtained by time lapse cinematography [5] of slow and fast dividing cell lines. D: Best fit of defined by Eq. 4 (solid line) to inter-mitotic time distribution density measured by long-term video tracking of in vitro proliferating B-cells [10]. The data in C and D were read from the graphs in the original publications ([5] and [10] respectively).

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(1)where is the reciprocal of the rate of the exponential (measured in units) and is the fixed delay (in units), and denotes the Heaviside step function whose value is zero for negative argument, i.e., for and one for positive argument. Notice that with a slight abuse of notation we denote here the random variable (subscript of density function ) and the value it assumes (the argument of the function ) by the same symbol This will allow us to denote the probability density function and the cumulative probability distribution of the random variable by and respectively, and to define the complementary cumulative distribution The delay in Eq. 1 ‘ensures’ that a cell that enters a specific phase will remain therein for at least time units (e.g. hours) before proceeding to the next phase. Besides this fixed minimal time additional less predictable effects that affect the completion of the processes associated to a phase are assumed to be exponentially distributed with both mean and standard deviation given by The phase specific mean completion time, denoted in the following by is then with standard deviation and coefficient of variation The Laplace transform of Eq. 1 is given by(2)

where is the transformed variable corresponding to the time lapse The temporal organization of the cell cycle is defined by the vector of phase-specific completion times, which in turn depend on the parameter vectors and The cell cycle length, understood as the time lapse between the entry into until exit out of is the random variable Its probability density function is the convolution of the three underlying delayed exponential distributions and corresponds to the delayed hypoexponential distribution. Explicit expressions can be computed using the inverse Laplace transform of the product of the Laplace transforms of the three densities given by Eq. 2, i.e.,(3)

In case that all entries in are distinct, we get(4)

in which the indices and iterate over the three phases and is the sum of the elements in

In Fig. 1 B we plot the shape of the phase specific completion time distribution defined by Eq. 1, which illustrates that the probability for a cell to complete a given phase in less than time units is zero under this model. A graphical representation of the cell cycle model is provided in Fig. 1 A. Notice that each phase can have distinct parameter values and for the completion time distribution.

As a first validation, we compared the empirical frequency of undivided cells as a function of time after ‘birth’ (reported by [5]) with the respective probability according to the model which we denote as (Fig. 1 C). As a second test, we fitted the cell cycle length density given by Eq. 4 to data extracted from video-tracking of in vitro proliferating B cells [10]. The delayed hypoexponential distribution (shown in Fig. 1 D), but also the delayed log-normal and the delayed gamma distribution (not shown) with parameter values proposed in [10], reproduce closely the measured division time histogram. While the two latter depend on three parameters each, the hypoexponential distribution depends on six parameters, that remain largely undetermined given this kind of data.

Balanced growth

A proliferating cell population that obeys the probability model specified in the previous section can be represented by a non-Markov multidimensional random process, whose evolution depends on its history. There exist an infinite number of possible histories or realizations of the population size dynamics We focus here on a specific important subset, namely those under balanced growth. Under balanced growth a cell population grows exponentially with mean growth rate and constant mean proportions of cells in the three phases where e.g., The expectation operator is defined over all possible realizations of the process.

We will now derive explicit expressions for and a transcendental equation that defines the growth rate. A first step in obtaining the constant frequencies of cells in each of the phases consist in computing the ratio between the cells that complete a given phase and the total number of cells inside the same phase at time This phase-specific quantity, denoted here by represents the asymptotic efflux rate constant, which will be useful, as we will see, to construct a transition probability matrix The latter will enable us to employ methods from linear algebra to solve the steady state condition.

Suppose for example that a cohort of cells entered a given phase at time Then the density of cells leaving this phase at time will be Similarly if a cohort of cells entered this phase at time then a proportion will remain in it until time

Recalling that the influx of cells into a given phase is proportional to and that is the complementary cumulative distribution of which is Laplace transformed to we integrate over all past entries and finally take the ratio to obtain(5)(6)(7)

While the second equality is a consequence of the definition of the Laplace transform, the third equality follows by substituting using Eq. 2. For a phase without a delay, i.e., the last expression simplifies to the familiar mass action principle, where the transition probability is directly proportional to the decay rate Assuming that cells are immortal and recalling that division occurs as cells proceed from to we build up the transition probability matrix as follows(8)

The balanced growth condition can now be formulated in matrix form(9)

where the growth rate is an eigenvalue of and the proportions vector is the corresponding eigenvector. It can be shown that there exists a single dominating real positive eigenvalue for (see Materials and Methods) whose associated normalized eigenvector is(10)

The uniqueness and existence of a dominating positive real root ultimately motivates our focus on balanced exponential growth, as any immortal proliferating cell population with sufficient nutrients and space will eventually enter this stationary phase. The time it takes, either starting with a single cell or a synchronized cell cohort to enter this state depends on the cell cycle parameters. The exponential growth rate is the unique real positive root of the characteristic equation which writes as(11)

It is easy to see that the denominator in Eq. 11 is always positive. To determine a non-trivial it remains to solve the transcendental equation in the numerator(12)

Numerical solutions to this equation can be computed using e.g., the Newton-Raphson root finding algorithm, with fast convergence if the initial value is set to where is the average cell cycle length, i.e., the sum of the elements in This first guess is a naive estimate for assuming that cells divide according to a deterministic division time identical to the average of the hypoexponential density defined in Eq. 3.

Learning from cell frequencies measured under balanced growth

The predicted fractions of cells in each of the phases can be compared to frequencies extracted experimentally from bivariate analysis of cell populations transiently exposed to nucleoside analogs and subsequently examined both for the intensities of the signals due to incorporated nucleoside analog and total DNA content [26] (e.g. the so called BrdU-DAPI staining dot plot). The question that we want to address in this section is: What can potentially be learned about the parameters of the model, given this type of experimental data? By definition, the measured frequencies will sum to one, and therefore we have for three populations effectively only two equations but six model parameters. This makes it impossible to identify all the parameter values, irrespective of the number of samples we take. It is however possible to derive analytical expressions for the upper and lower bounds for both the parameters and the average completion time of each phase.

Consider the experimentally determined frequencies, denoted by Substituting the vector by in Eq. 10 and solving for each phase specific parameter we obtain(13)

where is a phase specific element of the vector(14)

The phase specific parameters and respectively the reciprocal rate and delay, are by definition greater or equal to zero. These conditions propagate into Eq. 13 which allows us to specify boundaries for and First notice that is, for each phase, a monotonically decreasing function of with a maximum at and a zero crossing at The maximum and the root represent the upper bounds for and respectively, while the lower bounds are zero for both. We thus have for each phase(15)

The mean phase-specific completion time, the sum of the reciprocal rate and the delay is also bounded, with an interval given by(16)

This result is derived from the fact that is concave having its unique minimum at which follows from setting the derivative to zero. This implies that is a monotonically decreasing function in the interval with the corresponding extrema specified above. It is important to note that the intervals defined by Eqs 13–16 depend on the average growth rate which is in general not known. Formally if one specific pair of parameter vectors and explains the measured frequencies with growth rate the scaled parameter vectors and mimic equally well the same data for arbitrary positive however with a reduced growth rate This can be easily verified by substituting these expressions in Eq. 10 and Eq. 12. The direct consequence is that remains undefined. However for the relative average time a cells spends e.g., in phase the growth rate cancels out.

Using the fact that and the appropriate series expansion for the natural logarithm, the widths of the intervals bounding and for each phase can be written as:(17)

From this it is straight-forward to show that This implies that by using measurements of the phase-specific stationary cell frequencies to infer the phase-specific completion times results in estimates of the mean value that are more precise than the estimates of the standard deviation Notice that the width of the intervals can be interpreted as a naive lower bound for the uncertainty about the respective parameter values. For the two data sets analyzed in this article (see details in next section), we computed the intervals for the phase-specific standard deviations that were on average times wider than the intervals for the expected phase-specific completion times

Transient unbalanced growth

Balanced growth analysis does not allow to distinguish between fixed () and purely exponentially distributed () completion times even if is known. This follows from Eq. 15 because possible values for the standard deviation include and and the latter requires, according to Eq. 16, the delay to be null.

The incapacity to resolve the values of and is overcome if one selects and follows a subpopulation within which the proportions of cells in each phase are transiently different from the balanced growth proportions. Consider a simple thought experiment that consists in taking a population under balanced growth and labelling all the cells that are in a specific phase, say which can be either or Initially all the cells are in the same phase but as time passes by the labelled cells progress through the cell cycle and eventually distribute over the three phases. The labelled cell subpopulation which is initially not balanced will return asymptotically to balanced growth conditions, restoring the corresponding proportions of cells in the three phases. We refer to this transient dynamics of a selected subpopulation as transient unbalanced growth. It turns out that measuring the transient dynamics of this subpopulation yields information that potentially allows to distinguish between a fixed and a purely exponentially distributed phase completion time. More specifically, a mathematical proof will show that taking samples at three well chosen time points (support points) permits under ideal conditions accurate estimation of the average and the variability in the time required to complete the phase

The initial average fraction of cells in phase which are selectively labelled at time is determined by Eq. 10. To predict when the labelled cells will have completed we need to specify when they entered this phase. For the time before labelling the average influx into is proportional to . For the time after the labelling, because by definition all labelled cells entered phase before (otherwise they would not be labelled ‘as being in phase ’), the entry of cells is zero. Hence, the average influx to the labelled subpopulation is proportional to where denotes the Heaviside step function. Let us assume that within the subpopulation of labelled cells and their progeny one could identify how many phases a cell or a cohort of cells went through since the labelling event, and let count the number of phases since labelling.

In close analogy to expression Eq. 5 we compute the time-dependent exit-rate density distribution for cells with as(18)

where, for convenience, we interpreted and will interpret in the following both as a phase and a phase index. As before, the third row follows from the definition of the Laplace transform setting . On the left-hand side, the arrow from 0 to 1 represents the transition from the initial phase () to the next phase (), corresponding to the completion of the initial phase In contrast to Eq. 5, the denominator accounts for the cells that entered or initiated phase sometime in the past, and did not complete this phase until the instant of labelling (and not at time as in Eq. 5), while the numerator, except for the altered average influx, remains unchanged.

After computing and substituting using Eq. 2, Eq. 18 yields for (19)

It follows that the accumulated average cell flux that at time has completed and progressed to the next phase is given by(20)

which for approaches one, reflecting the fact that all cells will eventually complete

The Laplace transform of Eq. 20 writes as

where is, as before, the transformed variable corresponding to

Within a cohort of cells isolated for instance in phase, i.e., the accumulated average cell flux out of the subsequent phase can then be derived recalling Eq. 2 and using the properties of the inverse Laplace transform as(21)

For an arbitrary cell cohort originally in the accumulated average flux, completing phases and entering the phase since isolation, can be written in general as(22)

in which denotes a function which returns an appropriate phase index. For and it is defined as

where is the modulo operation, and is a vector of cell cycle phase indices. The function thus returns, for increasing , in a cyclical fashion, the cell cycle phase indices, starting with for Notice that Eq. 21 corresponds to Eq. 22 for and

Analytical expression for Eq. 22, although solved relatively easily with modern algebra software, can become quite cumbersome for values of larger than six. In our case, deriving the expressions for up to a value of five was sufficient to simulate the experiments.

Because we want to compare the model predictions with experimentally measured cell frequencies, more interesting than the accumulated fluxes are the expected proportions of cells inside each phase over time. These can be computed using Eqs 20–22, closely following the methodology outlined in [11], [12]. For the fraction of cells initially in phase we have(23)

where the lower index 0 in indicates that this expression describes cells which completed zero phases since The first term on the right hand side corresponds to the fraction of cells in phase at divided by which accounts for the total population growth during the same interval. The second term stands for the fraction of cells that remained in phase up to time relative to the initial number of cells in this phase. By evaluating the integral in Eq. 20, substituting in Eq. 23 and letting as before, without loss of generality, the time of partition be zero, we get for (24)

Expressions for cells initially in or phase can be obtained by substituting by the respective phase.

If there were no cell division (i.e., ) we could readily obtain the average fraction of cells that completed phases at time as the difference between the cells that entered the phase, i.e., , and those that left it, i.e., divided by To account for cell division, we need to multiply this difference by an additional term which increases by a factor 2 each time cell cohorts make a transition from This term is defined, for each case, as follows: and where the brackets in the exponent represent the floor operator.

In general we get for all consecutive phases for cells initially in phase the relatively manageable expression(25)

As for Eq. 24, the resulting solutions are defined as piecewise-continuous functions in time. Also notice that most expressions in this section can be written in more compact, but less intuitive, vector form, by dropping the initial phase index and using bold vector notation as before.

Learning from cell frequencies measured in transiently unbalanced growing subpopulations

In this section we will show that data from the transient kinetics generated by our thought experiment allows to accurately estimate the average and the variability in the individual completion times. The proof is based on the analytical expressions derived in the previous section, and also on the assumption that the kinetics are acquired under the ideal conditions of large population sizes and no measurement errors. The latter condition, although clearly unrealistic, can always be approached in practice by increasing the number of samples at each support point.

For the sake of generality, consider a subpopulation of cells that are in an arbitrary phase and are labelled at Assuming that the ‘label’ does not in any way affect the cell cycle of the cells, the parameters and of the labelled subpopulation are the same as those of the full population under balanced growth. Under these conditions, we can obtain using Eq. 13 and Eq. 14 with the fractions of the full population observed at time Substituting in the upper row of Eq. 24 and solving for to find(26)

where denotes an arbitrary time point that lies in the interval , and is the experimentally determined equivalent of Eq. 24. This shows that the balanced growth rate is fully determined by only two support points, one immediately after the partition at and a second at an arbitrary This also makes clear that placing more support points in the interval does not increase knowledge about nor the parameter values, under ideal conditions. Importantly the uncertainty about the phase-specific variability discussed in previous sections remains.

By replacing the same expression for in the second row of the right-hand side of Eq. 24 we get(27)

After experimentally acquiring and the phase specific and this expression will depend on a single unknown One can show that Eq. 27 is solved by a unique This follows from the fact that the right hand side of Eq. 27 is a monotonically decreasing function in with corresponding values lying in the interval while the left hand side is positive by definition. Substituting the solution for into Eq. 13 yields the remaining parameter vector

Taken together this proves that in theory samples of the three cell cohorts and taken at three support points, a first at a second at and a third at are sufficient to determine all the parameters of the model.

Conventional single pulse-labelling assays

The thought experiment analyzed so far, although conceptually simple, poses a series of experimental challenges, that make a one-to-one realization difficult. The technical difficulties lie mostly in initially separating the cells according to their phase and in following these cells as they enter the subsequent phases. A widely used technique, namely DNA-nucleoside-analog pulse-chase labelling experiments, generates nevertheless to a certain extent comparable data. The latter achieves the initial phase-specific partitioning by exposing during a short time window proliferating cells with a nucleoside analog (e.g., BrdU, IdU or EdU) that gets selectively incorporated into the DNA of cells that are actively replicating their genome. Measuring subsequently by simultaneously the DNA content and the amount of incorporated nucleoside analog per cell permits to discern the three phases and immediately after the pulse. In addition, due to the permanent staining property of the nucleoside analogs, it is possible to follow, up to a certain degree, the labelled and unlabelled cell cohorts over time. Several dies, such as Hoechst 33342, the dihydroanthraquinone analog DRAQ5, DAPI, and PI are commonly available to stain DNA content in cells [27], and can be used in combination with nucleotide analogs.

In theory, this method would largely correspond to the hypothetical experiment that we analyzed so far. In practice however, the overlap of the subpopulations in the scatter plots prevents the exact determination of the frequencies of cells described by Eq. 24 and Eq. 25. For example labelled cells that have completed the phase but remain in phase are indistinguishable from those that did not complete the initial phase yet. As has been reported previously, only four different sub-populations can be identified with reasonable accuracy [26]. These are:

• : labelled undivided cells which at time of labelling () were in phase ()
• : unlabelled cells that were in phase at ()
• : first generation progeny of labelled cells which were initially in phase ()
• : unlabelled cells and progeny of cells that were in at accompanied by the progeny of and ()

where the corresponding populations in our thought experiment are indicated in brackets. This shows that computing Eq. 25 up to is sufficient to describe a complete in silico BrdU pulse labelling experiment. The reason is that, using current protocols, fluorescence of labelled cells becomes indistinguishable from background as soon as the cells divide a second time. In other words, cells that leave population by dividing a second time join population (see Fig. 2). For the experimental data, analyzed in the next section, the fraction of labelled cells that completed two cell divisions during the 12 hours time frame of the experiment is negligible.

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Figure 2. DAPI-BrdU pulse-chase labelling FACS data.

Samples taken at several time points after pulse labelling proliferating U87 human glioblastoma cells with The four gated populations are and which are defined precisely in the main text. Briefly, the subscript indicates the phase at the instant of labelling, while the superscripts ‘u’, ‘lu’ and ‘ld’ refers to cells ‘unlabelled’, ‘labelled and undivided’ and ‘labelled and divided’, respectively. The data was generated as described in the Experimental Methods section.

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The population is the only sub-population that matches directly the type of data considered before and its temporal evolution follows as such Eq. 24. The remaining three populations in contrast represent mixtures of cell cohorts whose kinetics could be described individually by Eqs 24–25.

Learning from single pulse-labelling data

By analyzing two data sets from samples of single pulse-labelling experiments, we tested the model and the effect of population intermixing on the identification of the model parameter values. The two cell lines considered were in vitro cultured human glioblastoma cancer cells (for details see Materials and Methods) and in vitro cultured Chinese hamster cells (courtesy G. Wilson). We will refer to these data as the and the data sets. Both data sets consist of samples taken from asynchronously dividing cell populations at several time points after a single BrdU pulse, with sample sizes ranging from 5000 to 50000 cells each. Data points represent simultaneous measurements of BrdU as well as DAPI or PI (DNA content) in a single cell by fluorescent activated cell sorting.

As a preliminary test we minimized the residual sum of squares i.e., least-squares fitting, of adequate mixtures of Eq. 24 and Eq. 25 to extracted frequencies at different time points after the pulse. We found that, for properly chosen parameter values, both data sets were reasonably well approximated by the model predictions (Fig. 3 A).

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Figure 3. Model based parameter estimation.

A: Best fit of the model predictions (lines) to experimentally determined cell fractions after BrdU pulse labelling (dots). U87: In vitro cultured U87 human glioblastoma cancer cell line (three replicates). V79: In vitro cultured V79 Chinese hamster cells (single replicate) (courtesy G. Wilson). Best fit parameter values used to compute model predictions (U87: V79: units are hours). B: Approximate ML regions for the parameters and associated to each phase (gray: red: green: ). C: Bayesian bi-variate 99%-credibility regions for the parameters and for each phase. Arrows indicate point estimates and the dashed lines delineate the information that could have been gained in our thought experiment under noise-free conditions from two support points, one at and a second at . The U87 data set was generated as described in the Experimental Methods section. The V79 data set was a kind gift of G. Wilson.

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While this indicated that the model captured some of the relevant temporal characteristics of cell cycle progression, a subsequent analysis revealed that an infinite number of different parameter combinations fitted the measured frequencies with the same minimal (not shown). This implies that there exist, given the available data, no single best-fit parameter combination, but a whole region in parameter space that can explain the data equally well.

When we then interrogated the same data by approximate maximum likelihood (ML) estimation, using a simple likelihood function (see Materials and Methods), we found again that relative large regions in parameter space mapped to the same ML (see Fig. 3 B). It turned out that these regions were entirely superimposed onto the lines defined by Eq. 13 and Eq. 26 (dashed lines). These lines define what could have potentially been learned in our thought experiment with only two support points, one at and a second at . In both experiments, ML parameters associated with the phase were spread out almost everywhere along these lines (Fig. 3 B, gray regions). Parameters related to the phase were more concentrated but still in the case of the data a substantial region of ML estimates were observed. Finally the region for the phase parameters approached that of a point estimate for both data sets.

The spread of the ML estimates suggests that even in the ideal case of large population size and noise-free data, the specific choice of the support points in these experiments does not allow to determine uniquely neither the delay nor the standard deviation for all the phases. In contrast the average completion time for each phase and the total division time can be estimated with relatively high precision.

To better quantify the uncertainty of these estimates, Bayesian 99% credibility regions (CR) were computed by the Markov chain Monte Carlo method (MCMC) using the same likelihood function as before (Fig 3 C). CRs followed mainly the same trends as the regions observed in the ML estimates, covered however as expected a larger volume. An exception was the ‘blown up’ CR of the phase parameter for the cell line, for which the ML estimates wrongly insinuated a well defined point estimate.

In Table 1 we summarized the obtained Bayesian summary statistics. One can see that the intervals for the average duration of each phase are narrow compared to those for the individual parameters and . In both cases the data allows for a deterministic phase (), while for the data set variability in is a necessary characteristic to reproduce accurately the data. Notably, when contrasting the two cell lines, are the short phase of Chinese hamster cells and the approximately two times more extended phase of the human glioblastoma cell line. It is out of the scope of this paper to interpret or relate these differences to cell line specific conditions. More importantly in this context is the fact that the information of the analyzed data is too sparse to narrow down all the parameter values even under noise-free conditions.

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Table 1. Bayesian summary statistics.

https://doi.org/10.1371/journal.pcbi.1003616.t001

Redesigned dual pulse-labelling assay

The information extracted from the and data sets is apparently insufficient to pinpoint all six parameters related to the three phases of our simple cell cycle model. This is disappointing especially because the number of support points largely exceeds the three ideally required, and the support points seem to include at least for the U87 data set one at a second at and a third at

A potential explanation for this poor resolution in the estimates is the previously mentioned intermixing of the cell population clusters in the BrdU versus DAPI scatter plots compared to the ideal conditions discussed earlier. The cluster overlap in the data makes it impossible to measure directly the frequencies of most of the populations, including the cell cohorts described by Eq. 24.

In order to approach the conditions assumed in the thought experiment by avoiding the loss of information caused by the intermixing, we devised an extension of the current single pulse protocol, which places a second pulse immediately before measuring or fixing each sample (see Fig. 4, top). The second pulse is expected to expose the cells with a further nucleoside analog that can be distinguished from the first one by Depending on the cell cycle kinetics and the length of the measuring period, the additional pulse increases the number of classifiable populations from four up to nine distinct populations.

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Figure 4. Dual pulse protocol.

A: Simplified schematic representations of the protocols corresponding to a conventional single pulse labelling with one nucleoside analog (e.g., BrdU) and a dual pulse labelling experiment with two different nucleoside analogs (e.g., BrdU together IdU or EdU). B: Artificial staining of single-pulse labelling data (for original data see Fig. 2), showing eight of the nine subpopulations that could potentially be identified with double-pulse labelling. Notice that the four population and that can be followed by the conventional protocol, have each been subdivided according to the cell cycle phases. The naming convention for the populations is as follows: the superscript ( = ‘labelled undivided’,  = ‘labelled divided’,  = ‘unlabelled’) indicates whether the population is labelled and whether it has divided since the time of the first pulse; the first and the second subscript ( ) stand for the phase in which the population was at the time of the first and the second pulse respectively. Double subscripts are used only when necessary.

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To appreciate the additional populations identified by double pulse labelling, data from a single pulse-chase labelling experiment was artificially colored, to mimic the expected FACS output from proliferating cells labelled according to the protocol described before. In Fig. 4, besides the gates defining the populations and cells that have incorporated the second label are drawn in red. For the time immediately after the pulse (i.e., ), no extra information is gained by the second pulse. However, already two hours later, one additional population can be discerned. Twelve hours after the first pulse, seven population, instead of three, can be recognized. Thus by resolving the four initial population according to the cell cycle phases, it is possible to measure the kinetics of nine subpopulations ( and ). Because all these kinetics depend on the cell cycle parameters, each of them can in principle tell us something about the phase completions times. However some information is redundant. For example if and are measured, then is defined by the total fraction of cells in phase, because Similarly from one can deduce by knowing the frequency of cells in phase.

Double-label experiments using pairs of nucleoside analogs like BrdU, IdU and EdU, also in combination with radioactive tritiated thymidine (), have been explored in several cancer cell proliferation studies [19], [28]–[31]. In recent years, dual pulse experiments using BrdU in combination with EdU have become more common. Studies relying on this method estimated changes in replication, inferred mitochondrial DNA bio-genesis and stained proliferating cells in the bone marrow in vivo [32]–[34], in general with the aim to increase the statistical power of the conventional methods.

To assess if the latter method would allow quantifying more accurately and precisely the parameters of the model, we generated in silico data mimicking the output of a hypothetical dual pulse experiment using Eq. 24 and Eq. 25 (see Fig. 5 A). We found that by employing the redesigned protocol with the same replicates and time points as in the corresponding data sets, we could reduce the regions corresponding to the ML up to point estimates (Fig. 5 B). Furthermore, the uncertainties due to noise became also significantly smaller (Fig. 5 C). Pooling this artificial data according to the output expected from a single pulse experiment, reproduced again the uncertainties seen in Fig. 3 C (not shown). Together this indicates that the redesigned dual pulse protocol provides parameter estimates with higher accuracy and precision. Real dual pulse labelling experiments will however be needed to confirm these theoretical predictions.

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Figure 5. Analysis of simulated dual pulse labelling data.

A: Average kinetics of unlabelled (dashed line) and labelled cell cohorts (colored lines) were computed from Eq. 25, using ML parameter estimates from the U87 and the V79 data sets (U87: V79: units are hours). Support points and repeats were chosen according to the real experiments. Multinomial noise was added, mimicking the residuals found in the original data sets (see the Computational Methods section for more details). Finally, model solutions (lines) were fitted to the synthetic data sets (triangles). Best fit parameters (U87: V79: units are hours) B: ML parameter estimates from simulated data. All ML regions converge to point estimates (arrows). Squares indicate parameters used for generating the data (see A). C: Bayesian bi-variate 99%-credibility regions for the parameters and for each phase, based on the artificial data.

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Robustness of the estimates to other probability distributions of the phase completion times and to concurrent cell loss

The cell cycle model introduced here is deliberately simple and neglects cell loss. In this section, we ask whether the estimates of its parameters are reasonable when some of the simplifying assumptions of the model do not hold. Specifically, we ask how accurate are the mean and standard deviation of the phase completion times estimated using this simple model if the true completion times were not distributed as a delayed exponential function or if there was concurrent phase-specific cell loss.

Empirical measurements [35] indicate that the cycle phase time for the S phase is distributed closer to a delayed hypoexponential or a delayed gamma distribution (see below) rather than the caricatural delayed exponential. Therefore, an important question which arises is how much do the estimates of the average and standard deviation in phase durations obtained with this simple model depend on the true underlying distribution? While many different scenarios could be tested we opted to fit a delayed hypoexponential density with two decay and one delay parameter to direct in vitro measurements of and phase durations employing fluorescent biosensors (Fig. 6 A-B, [35]). Using the obtained best-fit estimates, we then performed in silico dual-pulse labelling experiments, in which the phase durations were drawn in the case of the and phase from delayed hypoexponential density functions (Fig. 6 C). Finally we fitted the simple model, i.e., Eq. 24 and Eq. 25, which is based on delayed exponential distributions, to this data, to see if we could recover the original averages and standard deviations despite using the ‘wrong’ caricatural model. Both summary statistics (i.e., mean, standard deviation) of phase durations could successfully be re-estimated (Fig. 6 D). Although generalizing this finding lies out of the scope of this article, it suggests that even if the true underlying distribution is not a delayed-exponential function, important quantities like the average and standard deviation of the phase durations may still be estimated with the simple model developed herein. It also indicates that BrdU labelling experiments with a realistic number of samples are unlikely to have the power to discriminate between delayed exponential and more complex density distributions.

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Figure 6. Robustness of parameter estimates to empirical phase duration distributions that are not delayed exponential functions.

A-B: Least-squares fitting of histograms predicted from a hypoexponential distribution with two decay and one delay parameter to measurements of phase durations using fluorescent biosensors [35]. The number of cells that were tracked in the original study was around 15 cells. C: Best fit of the cell cycle model with delayed exponential completion time distribution densities to synthetic data generated from a model with hypoexponential completion time distribution densities for the and phase with parameters as in A and B. D: Recovery of the initial distribution densities (solid lines) using the delayed exponential model (dashed line). Both the average and the variability in the phase completion time distribution (original average: 10.70 h, estimated average: 10.88 h; original std: 2.03 h, estimated std: 1.99 h) were estimated accurately. The data shown in A-B was read from the graphs in the original publication ([35]).

https://doi.org/10.1371/journal.pcbi.1003616.g006

We now turn to the issue of how much the presence of phase-specific cell death (or loss in general), which is unaccounted for in our model, affects the accuracy of the estimates of the mean and standard deviation of the phase durations. To this end, we will first introduce the extensions necessary to describe cell death in the model. We rely on the fact that if the probability of death per cell cycle is less that 50%, the average population size will asymptotically grow exponentially with an effective growth rate where This implies that the arguments used to analyze exponential growth without death remain valid for a model that allows moderate levels of cell death.

To consider death, we assume that cells have two possible fates per phase, either they progress to the next phase or they die. Let as before, be the phase completion time density, conditioned however on the cell being alive at time And let be the phase-specific time to death density conditioned on the cell having not progressed to the next phase. Then, as e.g., in [36], assuming that both events compete with each other (i.e., whatever fate happens first, prevents the other), the resulting density becomes(28)

Consider now a scenario of an exponentially growing population, in which cell death occurs exclusively during phase Let us assume further that the phase specific time to death density is a simple exponential density with mean Using straight-forward probabilistic arguments, we can compute analytically, for this simple scenario two important quantities, namely the probability to die in this phase (), and the expected value of the effective completion time, distributed as We get

Note that, in this simple case, is also the probability to die per division cycle.

Evaluating Eq. 18 using instead of we obtain for Eq. 24,(29)

where and represent the equivalents of and we had previously defined for the case of no cell loss. The former quantities, which now depend on are derived applying to Eq. 5 the same substitution as above. Expressions equivalent to Eq. 10 and Eq. 11 are obtained along the same lines. These become however rather lengthy and are therefore omitted here. Eq. 29 reproduces accurately in simulated BrdU pulse labelling experiments, if death occurs, as specified above (see Fig. 7 A for an example with and ). The differences between the analytical predictions for with 30% death and without death (denoted by ) are, for the parameter sets that we tested, relatively small, and vanish as expected, as tends to zero (see Fig. 7 B for computed at one specific time point ( h) for different values of ).

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Figure 7. Effect of cell death and completion time distribution on parameter estimates.

A: Comparison of analytical predictions (lines, Eq. 29) with simulated BrdU labelling experiment (squares). Cell death is assumed to occur exclusively during S phase with probability 0 (red) and 0.3 (blue) respectively. Only the population is considered. Parameters: units are hours. B: Difference between Eq. 29 (accounting for cell death) and Eq. 24 (neglecting cell death) at time h (see dashed line in A), as a function of C: BrdU labelling experiments were simulated assuming gamma distributed phase completion times (red curve, graphs on left column) and cell death during S phase with probability and (green curve, graphs on left column). The effective completion time (gray density plot, left column), the population growth (middle column) and the estimation of the mean and the standard deviation of are shown for both cases. Approximate confidence intervals for the estimates are computed as 1.96 times the standard error. Even though and the population growth are strongly influenced by the value of both and the estimates extracted from are barely affected. The dashed lines in the middle column indicate the time of the first pulse, which was chosen such that the average population was similar in both scenarios. Parameters for gamma distributed completion time distribution of the three phases: shape: scale: delay:

https://doi.org/10.1371/journal.pcbi.1003616.g007

To further test, how much both cell death and a completion time with a shape distinct from a delayed exponential may jointly affect parameter estimates, we simulated BrdU pulse labelling experiments, where two major assumptions underlying Eq. 24 were simultaneously violated. First, we assumed a delayed gamma distribution (with shape parameter of two) for the completion time of each phase. Second, we considered cell death during phase, and adjusted such that was either zero or The population size (starting with five cells) took about twice as much time to grow to a similar size for compared to a the scenario without death (see Fig. 7 C, middle column, for five independent simulations). In addition, the variability in the population sizes between the simulations appeared higher for increased death rates. In contrast, when estimating the mean and variance of by non-linear least squares fitting using Eq. 24, the marked changes seen in the population kinetics where not paralleled by changes in the estimates. Both the mean and the variance were accurately determined in both cases (see Fig. 7 C, right column). Taken together, this suggests, that the estimates for the mean and variance of using Eq. 24, at least for the reasonable parameter values that we tested, are relatively robust to simultaneous changes in the shape of the completion time, and moderate levels of cell death.

Stability analysis

Here we will show that a cell population that follows the stochastic model specified before will eventually enter a stationary exponential growth phase. The requirement for such an asymptotic behavior is, recalling Eq. 11, that the complex valued function(30)

has for positive valued elements of the vectors and a unique positive real root which represents the upper bound of the real part of any of its other potentially infinite number of roots. The complex numbers that solve Eq. 30 correspond, according to our model, to the stationary phase growth rate of the proliferating cell population. In case that is real, the population is growing exponentially, while if is purely imaginary growth is oscillating. In general, roots have both non-zero real and imaginary parts, which leads to oscillations with growing or decaying amplitude. If for real and we write the real and imaginary part of are computed as(31)

where(32)

For to be a root of both real and imaginary part have to vanish.We restrict our analysis to the positive complex half plane, i.e. since we are interested in growing and not contracting cell populations. Due to the symmetries in the trigonometric functions and and and one can easily see that if is a root, its complement is also a root. We can thus reduce the analysis even further to values with positive imaginary parts. If for fixed we plot in the complex plane as a parametric function of we get a spiral with the distance from a center point given by(33)

Crucially, as is a monotone increasing function of the spiral never crosses itself. For the imaginary part of vanishes as expected because and For this special case is obviously monotone decreasing with and restricted to the interval This means that the spiral can only ‘start’ in the interval between one and minus infinity. Taken together, this implies that if for and fixed the real part of is positive, then there exist a single ‘opportunity’ to cross the origin, while if negative there exists none. At the border where the real part is zero (Fig. 8 C), the corresponding value of is the only positive real root. Due to the monotonicity of any value of greater than the positive real root will result for in which does not admit for any solution. The different possible scenarios are exemplified in Fig. 8.

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Figure 8. Stability analysis.

as a function of for fixed values of For (green circle) the real part of Q takes, depending on a value in the interval The values for x are increasing from A-D, while and remain unchanged. For relatively low values of (A-B) the real part is positive for After one or several turns, i.e by increasing the spiral can potentially cross the origin only once (empty circle). In A the spiral misses the origin, while in B the spiral crosses the origin after one turn. Crossing of the origin means that the corresponding complex number is a root of Q. In C the spiral starts at the origin. This represents the only real positive root of Q. For initially negative values of (D) the spiral can never cross the origin because the distance to the center point (gray circle) is already in the beginning for larger than the distance between the latter and the origin. By increasing y this distance will even grow further according to Eq. 33.

https://doi.org/10.1371/journal.pcbi.1003616.g008

Experimental methods

Computational methods