Cell cycle kinetics in the pedigree as bifurcating autoregressive process.

Time-lapse microscopy allows tracking of individual cells and identifies family relations among cells (Fig 1A). Using this information we investigated the inherited and environmental causes of cell cycle (CC), cell-cycle phases length (G1 and S/G2/M phases) and protein dynamics variability. We observed strong positive correlation of cell cycle between siblings, whereas the same correlation for parent and progeny (θ) was significantly smaller (Fig 1B). Correlation between cell-cycle phases follow the same pattern, but the correlation between durations of G1 phases is always lower than correlation between durations of S/G2/M.

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Fig 1. Family relations among cells observed in vitro.

(A)Time-lapse microscopy permits the tracking of individual cells, identifying division moments, family relations, cell cycle and cell-cycle phase duration, and protein concentration at the time of each observation. Introduced notations and colors indicate cell cycle (CC, blue), G1 phase (G1, red) and S/G2/M phases (green). Corresponding simulation data are represented by the gray color. (B) Correlations between family members based on experimental and simulation data. Estimation of standard deviations from experimental data is described in detail in Supporting Information (S1 Text, Methods). Simulation results for the autoregression bifurcation model are described in detail in Methods.

https://doi.org/10.1371/journal.pcbi.1007054.g001

Our results suggest that progeny cells inherit properties from their parents (Fig 1B). Consistent with this, we observed even stronger correlations between siblings. In the literature, such correlations are frequently explained by external factors affecting the cell-cycle length, such as environmental conditions, among-cell communication, neighborhood effect or the age of cells (see ref. [25] for the NIH 3T3 cells). In our data, after two generations the correlation disappears, as shown by low-correlation coefficients between grandparent and grand-progeny cells (S3A Fig).

As is later demonstrated, experimental results are consistent with the bifurcating autoregression model, in which the progeny cells inherit random but correlated fractions of a given feature of the parent cell [26]. Bifurcating autoregression has been successfully used previously to fit cell-cycle kinetics data recorded as magnetic tape videos [27]. The approach presented in the current paper is more complex; it includes modeling of phases of cell cycle and protein concentration and is based on a significantly larger dataset.

Analysis of cell-cycle kinetics in the pedigree, with results for the third generation of cells (cousins, grandparent—grand-progeny), including the physical proximity and cell birth time which do not affect cell-cycle duration, are presented in Supporting Information (S1 Text, Hereditary and environmental causes of cell cycle length variability based on the 15% FBS experiment).

Modelling the pattern of dependence between G1 and S/G2/M phases of cell cycle.

Previously [3], the bifurcating autoregression model was applied to cell-cycle duration data, but not to dependence between durations of cell-cycle phases. We investigated the statistical dependence of cell-cycle phase durations and overall cell-cycle times, and found a strong positive correlation between durations of cell-cycle phases and cell-cycle times (Fig 2B). However, there exists only a low correlation between durations of G1 and S/G2/M phase length (Fig 2B). Based on detailed analysis of presented results, including clustering of cells with different progression rates in G1 and S/G1/M (Supporting Information, S1 Text, Statistical dependence of cell cycle phase durations and cell cycle times), we decided that inheritance of the durations of the phases of the cell cycle can be modelled independently. Accordingly, we used the bifurcation autoregression model to estimate the durations of the G1 and S/G2/M phases in related cells (cf. schematics of the model in Fig 2A., and Estimation of Parameters using the method of moments in Methods). Patterns in the experimental data are well reproduced by the results of modelling. In Fig 2C, we presented the distributions of the experimental and simulated data, and in Fig 2B the noise-perturbed linear relationships between the total division time and the durations of the G1 and combined S/G2/M phases in simulation data. Results confirm that we can treat durations of cell-cycle phases independently, which implies that lengthening one phase does not imply lengthening of other phases.

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Fig 2. Analysis of protein dynamics based on experimental and simulation data.

(A) Schematics of the model. In the population, each cell is characterized by seven parameters. At division, the parent-cell mass (C_end, G_end) is randomly split between the two progeny cells, according to the expression in which β is the random variable. Cell-cycle phase durations () for progeny cells (i is progeny number) are calculated using autoregression bifurcating models. —production rate of Cdt1 and Geminin proteins, are calculated using linear regression models, —cell cycle duration, T_G1,T_SG2M—G1 or S/G2/M phases length of the parent cell, μ_G1,μ_SG2M–mean G1 or S/G2/M cell cycle phases length, θ_G1,θ_SG2M–relation between parent and progeny G1 or S/G2/M cell cycle phases length, – random variables from bivariate lognormal distributions (common mean zero, common variances and correlation coefficients). We abandoned the originally assumed bivariate normal distribution to lognormal because of the positive skewness of the distributions of cell-cycle duration. (B) Comparison of linear relationships between the total division time and the duration of phases for experimental and simulation data. Solid black lines show the fitted linear relations of the form y = (slope) × x. (C) Comparison of distributions of the cell cycle and the cell-cycle phase duration for experimental data and modelling results. (D) Looking for probable regulatory mechanisms. Data-derived and simulation-based correlations between pairs of variables characterizing the protein trajectories. (E) Data-derived and simulation-based correlations between production parameters of family members.

https://doi.org/10.1371/journal.pcbi.1007054.g002

Modelling protein dynamics in individual cells.

As we described in ref. [4], we fitted experimental trajectories of individual cells to a model in the form of a system of ordinary differential equations (ODE) and calculated correlations among production parameters, duration of phases of cell cycle and initial levels of proteins. Each cell in the population is described by seven parameters with cell-dependent stochastic values. Based on the matrix of correlations presented in Fig 2D (left part) we proposed a procedure described in the Methods section and in Fig 2A, with the added noise being correlated between siblings. Correlated noise reproduces weak correlation of protein dynamics between parent and progeny, and strong correlation of siblings evident from correlation matrices of protein levels in related and unrelated cells (diagonal area, S5 Fig). Comparison of the correlation between protein production parameters assumed in the simulations and those recovered directly from the experimental data (see Methods, Estimation of parameters) is presented in Fig 2E. Detailed scatterplots of experimental and simulated data (S11 Fig) demonstrate the accuracy of the model. Simulation results reproduced most of characteristics observed in experimental data (Figs 1–3).

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Fig 3. Results of long term simulations.

(A) Dynamics of the time series of the Cdt1 and Geminin protein contents in a randomly chosen lineage of descendants of the ancestor cell. (B) Phase portrait of experimental and simulation data. Ten trajectories of cell cycle were randomly selected from the experiment (left) and simulation (right) data. In both cases, fluorescence levels were normalized to maximum values.

https://doi.org/10.1371/journal.pcbi.1007054.g003

Long-term simulations: Memory and transients.

We have run the model for a large number of cell generations (50 generations), recording the protein contents in cells of the resulting pedigree at constant time intervals. Simulation was started from a single ancestor cell with defined or randomly sampled parameters, and the fates of cells from the resulting populations have been recorded. During cell division, the proteins are unequally distributed among progeny cells [4]. Direct measurement of the extent to which division of proteins is unequal is difficult due to degradation of the FUCCI markers in late M phase. We performed simulations to calibrate the impact of this distribution’s width on the dynamics of protein levels in the progeny cells. Strong asymmetry in protein division does not affect the regulation mechanisms modelled, and therefore we used the parameters proposed in ref. [4], calculated for mammalian cells. A sample long-term simulation and resulting phase portraits are presented in Fig 3. Stochasticity of cell-cycle duration and protein production parameters result in large variability in protein expression, attenuated by regulatory mechanisms.

Phase portraits are the usual way of depicting systems dynamics in general [28]. Fig 3B depicts phase portraits based on experimental data (left) and on simulation results (right), which coincide quantitatively for most observed cell trajectories, with the qualitative pattern being similar in almost all cells. Discrepancies are likely to be due to high level of measurement-related noise in the cells followed experimentally. An example of such “noisy” measurement is presented in S12 Fig

We designed long-term experiments to verify if the initial ancestor cell has an impact on cell cycle durations in later generations. Our first experiment was based on simulations for 400 generations of a randomly chosen line of descent originating from a single cell. Histories of cell-cycle times for single cells showed stabilization properties (S8 Fig). The influence of the ancestral cell-cycle time became dissolved in subsequent generations.

Our second experiment was based on simulations of a growing population, started, as before, from single ancestor cells with different cell-cycle lengths. In two extreme cases (initial cell-cycle time 13.6 h and 61.3 h), large differences in the population growth rate were observed. Within the interval from 0 to 200 h, during which the cell-cycle duration in both populations returned to the stationary distributions, the descendants of the cell with the short cell-cycle length have formed a subpopulation 40 times larger than the descendants of the cell with the long cell-cycle duration. If these two subpopulations were pooled, the one originating from the ancestor with the shorter cell-cycle length would dominate the other.

Epigenetic drift in proliferating cell population. The effect dominance has on the descendants of “fastest cycling” cells, described in the last paragraph of the previous section, is not deterministic (see further on). Our results show that it is similar to genetic drift, and we use a working term “epigenetic drift”. The classical form of genetic drift used in population genetics [29], is embodied by the Wright-Fisher model (Fig 4A), which assumes a time constant population size equal to N and discrete generations. Each (n+1)-st generation individual is a progeny of a randomly chosen individual of the n-th generation. As a result, the number of progeny of any individual is binomially distributed, and the random vector X of progeny counts of all N individuals has multinomial distribution, i.e. X~MN(N⁻¹,…,N⁻¹;N). Eventually, for time n large and inversely related to N, the population is dominated by the progeny of a single individual (effect known as “fixation”). It is also known [29] that if the population size N = N(n) varies with n, then the Wright-Fisher model including fixation holds approximately, with “effective” population size being equal to the harmonic mean of N(1),N(2),…,N(n). Our simulation results show similar effects, however, with effective population sizes deviating from simple harmonic means (also, see Methods).

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Fig 4. Comparison of simulation results obtained by proposed model and Wright-Fisher model.

(A) Main assumptions of Wright-Fisher model and our model are presented in separate boxes. The most important difference concerns population size, constant in Wright-Fisher and variable in our model. Series of in silico experiments were performed with different initial population sizes of 3, 25, and 100 cells drawn from previously generated populations. Each cell was characterized by different cell-cycle time and at the time 0 cells were not synchronized (i.e. cells were spread over different cell-cycle phases). (B–D) Example of performed simulations for initial population with N = 3. B) Descendants of ancestor cells are identified and counted. Total population size is marked by black dashed line. C) The fractions of the progeny in population were calculated. We analyzed the fraction of progeny in the population at time 300, but the level is determined after t = 200. Simulations were repeated until required sample size was obtained. D) Cumulative distribution functions for simulation data and Wright-Fisher model. (E) Comparison of simulation data and Wright-Fisher model for three different values of initial cell count N. Histograms of simulation datasets (left) and random numbers drawn from estimated binomial distribution with K (right) represent the fractions of progeny in population after 300 h of simulations. Cumulative distribution functions for both cases are also presented to compare the tails of distributions.

https://doi.org/10.1371/journal.pcbi.1007054.g004

Fig 4 contains a summary of hypotheses and results. Assumptions of the Wright-Fisher model and our model are presented in separate boxes (Fig 4A). The most important difference concerns population size, constant in Wright-Fisher model and variable in ours. In Fig 4B–4E, the outcomes of a series of in silico experiments, performed with different initial population sizes, are shown. In these simulation runs, 100, 25, 10 or 3 cells were drawn from previously generated populations. Each cell had different cell-cycle length and at t = 0 cells were not synchronized (cells were in different cell-cycle phases). The number of descendants for each initial cell was determined (marked by distinct colors in Fig 4B, for the case N = 3). Total population size was marked by a black dashed line (Fig 4B). The fraction of each progeny in the population was calculated (colors in Fig 4C correspond to colors in Fig 4B). We analyzed the fraction of descendants in the population at time 300, while in general this fraction stays constant already after t = 200. Simulations were repeated until appropriate sample size was obtained. Outcomes are presented in histograms in Fig 4E; the mean value is specified for each histogram. Random numbers drawn from estimated binomial distribution with coefficient K values as specified in the legend are presented in Fig 4E. Distribution tail functions compared to the binomial are presented in Fig 4E.

Based on the Wilcoxon’s test, there is no significant difference between the normalized frequencies and the binomial distribution. The estimates of covariances (Table 1) are not equal to theoretical expected values, although they are of similar order, except for Dataset 1, which includes only 3 ancestral cells. Correlation coefficients have very skewed distributions [30], which can be one of the reasons for the discrepancy. Another possible reason is deviation from multinomial sampling. Wright-Fisher model is a special case of the so-called Cannings model, which allows more general (not necessarily multinomial) sampling distributions, and may result in different correlation coefficients. Cannings model is not frequently used, however it was applied to model stem cells in human colon [31]. Based on current data, it seems difficult to determine which of these two models might be more appropriate.

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Table 1. Comparison of data-estimated and Wright-Fisher model-predicted parameter values, variances, and correlation coefficients, based on simulated genealogies of proliferating cells.

https://doi.org/10.1371/journal.pcbi.1007054.t001

Heritable cell cycle phase times.

Residence times in progeny G1 and S/G2/M cell cycle phases, given the parents’ residence times, follow the bifurcating autoregression equations (1) (2) where the superscript i = 1,2 denotes the number of progeny cell; are G1 or S/G2/M phases length of progeny cell i, T_G1, T_SG2M are G1 or S/G2/M phases length of the parent cell. Constants m_G1,m_SG2M are equal to mean G1 or S/G2/M cell-cycle phase durations plus a corrective term required to ensure long-term equilibrium (see further on); θ_G1,θ_SG2M are fractions of the durations of the G1 or S/G2/M phases inherited from the parent; and the noise terms are random vectors from exchangeable bivariate lognormal distributions (component-wise transformation by exponential function of bivariate normal with equal means and variances). Cell-cycle time is the sum of residence times in G1 and SG2M

This model deviates from the original bifurcating autoregression of Cowan and Staudte which assumes bivariate normal distribution. The modification accommodates the nonnegativity and positive skewness of the distributions of cell-cycle phase durations.

Unequal division of FUCCI proteins between progeny cells.

We used the following stochastic multiplicative laws where C,C¹,C²,G,G¹,G² are the parent, progeny 1 and progeny 2 amounts of Cdt1 and Geminin, respectively, and independent random variables X_C X_G have beta distributions B(α,β), such that E(X_C) = E(X_G) = 0.5 and cv(X_C) = cv(X_G) = cv. Value of cv has been found based on data as explained later. This results in α = β = (cv⁻²−1)/2, considering that with α = β, cv = (2α+1)^−0.5. Beta distribution is used because it constitutes a flexible family of random variables restricted to the unit interval of the real line.

FUCCI Protein synthesis (coefficients based on linear regression, using initial concentrations of proteins and cell cycle phase lengths).

In normal cells, Cdt1 and its inhibitor Geminin play a role in cell-cycle regulation (S13 Fig), and one can indirectly affect the dynamics of the other. However, our measurements are based on fluorescence of markers containing only fragments of Cdt1 and Geminin. As a consequence, the markers are produced and degraded in proportion to the “full-scale” Cdt1 and Geminin, but they do not interact with other proteins [14, 15]. Single-cell dynamics of the concentrations of the two markers are described using the following equations (as earlier on, C variable for Cdt1, G for Geminin): (3) where following variables are used to describe the two phases:

p_C—production rate of Cdt1 protein, constant in time,

p_G—production rate of Geminin protein, constant in time,

d_C–background degradation rate of Cdt1 protein, constant in time,

d_G–background degradation rate of Geminin protein, constant in time,

d_C,SG2M(t) = a(t−T_G1)1(t−T_G1)—degradation rate of Cdt1 protein, linearly increasing in time during the G/S2/M phases of the cell cycle and equal to 0 otherwise,

d_G,G1(t) = b(T_G1−t)1(T_G1−t)—degradation rate of Geminin protein, linearly decreasing in time during the G1 phase of the cell cycle and equal to 0 otherwise.

The phase-dependent degradation rates were assumed to provide a simplified description of a large number of regulatory processes inside cells, such as formation of complexes, their activation and deactivation. As it is known, the cell cycle is controlled by ubiquitin (Ub)-mediated proteolysis. Activity of two complexes: APC^Cdh1 and SCF^Skp2 oscillate reciprocally during the cell cycle. Cdt1 and Geminin are direct substrates of these complexes [14].

For purposes of long-term simulations, the protein production rates of simulated cells are drawn from linear regression equations, following the assumptions that:

1. p_C depends linearly on T_G1 and C₀, and
2. p_G depends linearly on T_SG2M and G₀.

Mathematically, (4) where C₀ and G₀ are the amounts of the Cdt1 and Geminin, respectively, at the beginning of the cell cycle, and ε_G1 and ε_SG2M are random error terms. Random numbers are drawn from two-dimensional normal distribution to reproduce correlation between siblings: . Parameters were estimated using linear regression from single-cell data. There is generally high agreement among parameters simulated in this manner, and the single-cell data is depicted in Fig 2E.

Estimation of the bifurcating autoregression parameters.

We accomplish this task using the method of moments; in this case, the first and second moments. For clarity, we omit the subscripts corresponding to cell cycle phases G1 and SG2M. It is understood that estimation is carried out separately for G1 and SG2M.

The noise terms in Eqs 1 and 2 are distributed according to the version of exchangeable bivariate lognormal distribution, which arises when coordinates of an exchangeable bivariate normal vector X are transformed by the exponential function. Suppose X = (X¹,X²)~MVN(μ,Σ), where μ = (μ,μ) is the vector of expected values and is the variance-covariance matrix of rv X. The resulting bivariate lognormal rv W = (W¹,W²) = (exp(X¹),exp(X²)) has a joint density function that can be derived in a standard way, and the first- and second-order moments have the form hence the correlation coefficient is equal to

We assume now that

1. Equilibrium distribution arises in the long-time limit in the model, so that times T and T_i are all identically distributed. This is the consequence of the general theorem on time series models, see eg. the monograph ([58], Theorem 4.3.1., p. 304).
2. Cell measurements represent such equilibrium,
3. In the bifurcating autoregression right-hand sides, Wⁱ and Tⁱ are independent rv’s.

This allows deriving the following moment equations

Remembering that under equilibrium, E(Tⁱ) = E(T) and Var(Tⁱ) = Var(T), we obtain (5) (6) (7) (8) and combining expressions for parent-progeny and progeny-progeny covariances, we obtain cousin-cousin correlation coefficient (9)

We can now solve these equations for the four unknown parameters θ,σ²,μ, and m, assuming a value for parameter ρ. Following this, we can substitute for the moments E(T) and Var(T) and correlation coefficients and their sample-based estimates and , and obtain the method-of-moments estimates of the unknown parameters, given assumed ρ. Subsequently, we can adjust ρ so that the distributions of residence times T fit data best.

We first note that by Eq (8)

Now, we use Eq (7) in the form to write a transcendental equation for

Since , therefore, for ρ>A, there exists x = x₀>1, which satisfies this equation. This provides

The remaining two estimates are obtained by simple inversion of Eqs (5) and (6)

Estimation of the parameters of the distribution of unequal division of FUCCI proteins between progeny cells.

Value of the coefficient of variation cv of the random variable X in the multiplicative model of unequal division has been found, based on comparison with simulations. Direct measurement of the extent to which the proteins are unequally distributed between progeny cells is difficult due to degradation of the FUCCI markers in late M phase. We performed simulations to calibrate the impact of spread of this distribution on dynamics of protein levels in the progeny cells. We assumed the cv = 0.13 the same as that in ref. [4], calculated for mammalian cells. Varying cv did not alter the outcome in a substantial manner, therefore cv = 0.13 has been kept, which corresponds to parameter values α = β = 29.3 of the beta distribution model.

Estimation of FUCCI protein synthesis ODE model.

It is assumed that dynamics of protein synthesis can be described using Eq (3). Experimental trajectories of protein levels in individual cells have been used to find individual production rates, with degradation rates assumed the same in all cells. These and other parameters such as C_G1, the amount of Cdt1 protein at the end of G1 phase, and G_end, the amount of Geminin protein at the end of the cell cycle, were fitted iteratively using Eq (4) and the following heuristic relationships

Empirical parameters ϑ_C and ϑ_G link protein production rates with the ratios of the protein peak values and the respective durations of cell-cycle phases. This amounts to linear interpolation of the arcs of exponential functions.

Cell-cycle phase durations for all cells have been estimated using an original technique (S14 Fig) based on information about division times of the cells and their parents and fluorescence levels of FUCCI proteins. It allows detecting cell cycle and phases endpoints, but also verifying if the trajectory is of appropriate quality and can be subject of further analysis. Briefly, it includes the following steps (S14 Fig): (i) determination of the level of noise and of the parameter values for smoothing, (ii) computing of numerical derivatives of the trajectories of Cdt1, Geminin protein levels, (iii) detection of local minima of differentiated data to identify division times, and (iv) detection of Cdt1 protein maxima, the timing of which provides the estimated moment of transition from G1 to S phase of cell cycle (in this step we analyze only the interval of Cdt1 protein trajectory located between division times). The maxima of the expression of the Cdt1 protein, just before a rapid drop, are detectable in 99.7% of cells. In general, problems at each step of analysis were mostly caused by aberrant protein trajectories in some cells, such as the protein level not changing in time, or the almost constant noise level being extremely high, which made it impossible to retrieve the protein dynamics signal.

Application of the procedure resulted in the following estimates of parameters that are identical for all cells, d_C = 0.08, d_C−SG2M = 0.088, ϑ_C = 6, d_G = 0.0,d_G−G1 = 0.124, ϑ = 4.

Mathematical framework.

In the canonical Wright-Fisher model of population genetics, time is measured in discrete units (generations), and the number of individuals in each generation is constant and equal to N. Individuals of generation t contribute to the t+1-st generation by multinomial sampling. In mathematical terms, let us denote X_i the number of the t+1-st generation descendants of the i-th individual of the t-th generation. Then X = (X₁,X₂,…,X_N)~MN(N⁻¹,…,N⁻¹;N), which, among other things, implies that X_i~Binom(N⁻¹;N). The cell proliferation process we consider runs in continuous time and the cell count changes with time. Let us consider a population started at time t = 0 by N cells. Let us consider a time interval of length Δt and denote Z_i(Δt) = #{progeny of i at time Δt}. If the distribution of vector X = [Z(Δt)/|Z(Δt)|] ∙ K approximates a multinomial distribution, then we may consider the cell proliferation process as approximately conforming to a Wright-Fisher model with effective population size K (which depends on Δt). Technically, X is also a discrete random vector, although not integer-valued. However, comparisons based on distribution tails are valid for any random variables.

The rationale for our approach follows from practice well established in population genetics, of approximating real-life populations by populations obeying Wright-Fisher model with “effective population size” that is different from the actual “census population size”. For example, if the population size has been varying in the past, the effective population size is frequently approximated as a harmonic mean of the census population sizes in successive past generations ([29], Equ. (2.13)).

In our notation, this harmonic-mean estimate of K has the form of

Assuming that Z_i(t) are equal to their expected values, i.e. that Z_i(t) = Z_i(0)e^αt, where α is a growth rate estimated from simulated trajectories (for example, by representing them in semi-logarithmic scale and estimating the slope α of the graph using linear regression), we obtain

We will see that these two estimates provide almost identical values of K.

However, it has been established at least since the well-known publication of Rogers and Harpending (1992) [59] that the Wright-Fisher model does not generally apply with a good approximation when population undergoes demographic change. Nevertheless, we demonstrate here that the approximation is acceptable at least in the terms of marginal distributions (see Results). Explanation of possible reasons for this is postponed to Discussion.

Statistical considerations.

To obtain an informal test, we consider the marginal distributions, which under multinomial joint distributions are binomial, and the second moments. As stated, we expect that

Upon rescaling by the effective population size K so that Y = X/K,

From simulations, we obtain M sample trajectories generated by our model, which can be rescaled by the estimated effective population size. This latter can be accomplished by fitting the empirical distributions of X_i to the Binom(N⁻¹;K) distribution. As a result, we obtain a sample in the form of a table: . We consider two statistics and , where . Then, remembering that the entries of Y^(m) are not independent, we obtain

Using these two expressions, we can estimate by the method of moments, the variance and covariance under exchangeability assumption. The results are depicted in Table 1.