Hidden longrange memories of growth and cycle speed correlate cell cycles in lineage trees
Abstract
Cell heterogeneity may be caused by stochastic or deterministic effects. The inheritance of regulators through cell division is a key deterministic force, but identifying inheritance effects in a systematic manner has been challenging. Here, we measure and analyze cell cycles in deep lineage trees of human cancer cells and mouse embryonic stem cells and develop a statistical framework to infer underlying rules of inheritance. The observed longrange intragenerational correlations in cellcycle duration, up to second cousins, seem paradoxical because ancestral correlations decay rapidly. However, this correlation pattern is naturally explained by the inheritance of both cell size and cellcycle speed over several generations, provided that cell growth and division are coupled through a minimumsize checkpoint. This model correctly predicts the effects of inhibiting cell growth or cycle progression. In sum, we show how fluctuations of cell cycles across lineage trees help in understanding the coordination of cell growth and division.
Introduction
Cells of the same type growing in homogeneous conditions often have highly heterogeneous cycle lengths (Smith and Martin, 1973). The minimal duration of the cell cycle will be determined by the maximal cellular growth rate in a given condition (Kafri et al., 2016). However, many cells, in particular, in multicellular organisms, do not grow at maximum rate, and their cycle length appears to be set by the progression of regulatory machinery through a series of checkpoints (Novak et al., 2007). While much is known about the molecular mechanisms of cellcycle regulation, we have little quantitative understanding of the mechanisms that control duration and variability of the cell cycle.
Recently, extensive livecell imaging data of cell lineages have become available, characterizing, for example, lymphocyte activation (Mitchell et al., 2018; Duffy et al., 2012; Hawkins et al., 2009), stem cell dynamics (Filipczyk et al., 2015), cancer cell proliferation (Spencer et al., 2013; Barr et al., 2017; Ryl et al., 2017), or nematode development (Du et al., 2015). Such studies across many cell types have found that cycle lengths are similar in sister cells, which may be due to the inheritance of molecular regulators across mitosis (Spencer et al., 2013; Mitchell et al., 2018; Yang et al., 2017; Barr et al., 2017; Arora et al., 2017). By contrast, ancestral correlations in cycle length fade rapidly, often disappearing between grandmother and granddaughter cells, or already between mother and daughter cells.
Remarkably, however, the cycle lengths of cousin cells are found to be correlated, indicating that the grandmothers exert concealed effects through at least two generations. High intragenerational correlations in the face of weak ancestral correlations have been observed in cells as diverse as bacteria (Powell, 1958), cyanobacteria (Yang et al., 2010), lymphocytes (Markham et al., 2010) and mammalian cancer cells (Staudte et al., 1984; Sandler et al., 2015; Chakrabarti et al., 2018). The ubiquity of this puzzling phenomenon suggests that it may help reveal basic principles that control cellcycle duration.
Theoretical work has shown that more than one heritable factor is required to generate the observed cellcycle correlations in T cell lineage trees, while the nature of these heritable factors has remained unclear (Markham et al., 2010). Stimulated by the idea of circadian gating of the cell cycle in cyanobacteria (Mori et al., 1996; Yang et al., 2010), recent comprehensive analyses of cell lineage trees across different species have proposed circadian clock control as a source of cellcycle variability that can produce the observed high intragenerational correlations (Sandler et al., 2015; Mosheiff et al., 2018; Martins et al., 2018; Py et al., 2019); such a model also reproduced observed cycle correlations in colon cancer cells during chemotherapy (Chakrabarti et al., 2018). However, in proliferating mammalian cells in culture, the circadian clock has been found to be entrained by the cell cycle (Bieler et al., 2014; Feillet et al., 2014). Moreover, the circadian clock is strongly damped or even abrogated by oncogenes such as MYC (Altman et al., 2015; Shostak et al., 2016) yet MYCdriven cancer cells retain high intragenerational correlations (Ryl et al., 2017).
Ultimately, the cell cycle must coordinate growth and division in order to maintain a welldefined cell size over many generations. Yeast species have long served as model systems. Here, it is assumed that growth drives cellcycle progression, although molecular mechanisms of size sensing remain controversial (Facchetti et al., 2017; Schmoller and Skotheim, 2015). By contrast, animal cells can grow very large without dividing (Conlon and Raff, 2003), and recent precise measurements suggest that growth control involves both modulation of growth rate and cellcycle length (Sung et al., 2013; Tzur et al., 2009; Cadart et al., 2018; Ginzberg et al., 2018; Liu et al., 2018). A minimal requirement for maintaining cell size is that cells reach a critical size before dividing, which can be achieved by delaying S phase (Shields et al., 1978).
Here, we present a systematic approach to learning mechanisms from measured correlation patterns of cell cycles in deep lineage trees. First, we develop an unbiased statistical framework to identify the minimal model capable of accounting for our experimental data. We then propose a biological realization of this abstract model based on growth, inheritance and a size checkpoint, and experimentally test specific predictions of the biological model.
Results
Lineage trees exhibit extended intragenerational correlations
To study how far intragenerational cellcycle correlations extend within cell pedigrees, we generated extensive lineage trees by imaging and tracking TET21N neuroblastoma cells for up to ten generations during exponential growth (Figure 1A, Figure 1—video 1, Figure 1—source data 1 and Figure 1—figure supplement 1A). Autonomous cycling of these cells is controlled by ectopic expression of the MYCfamily oncogene MYCN, overcoming the restriction point and thus mimicking the presence of mitogenic stimuli (Ryl et al., 2017). High MYCN also downregulated circadian clock genes (Figure 1—figure supplement 2). The distribution of cycle lengths (Figure 1B and Figure 1—figure supplement 1B) was constant throughout the experiment (Figure 1C and Figure 1—figure supplement 1C) and similar across lineages (Figure 1—figure supplement 1D), showing absence of experimental drift and of strong founder cell effects, respectively. To determine cyclelength correlations without censoring bias caused by finite observation time (Figure 1—figure supplement 3A; Sandler et al., 2015), we truncated all trees after the last generation completed by the vast majority (>95%) of lineages. The resulting trees were 5–7 generations deep, enabling us to reliably calculate Spearman rank correlations between relatives up to second cousins (Figure 1D,E and Figure 1—figure supplement 3B).
Cyclelength correlations of cells with their ancestors decreased rapidly with each generation (Figure 1E). However, the correlations increased again when moving down from ancestors along sidebranches—from the grandmother toward the first cousins and also from the greatgrandmother toward the second cousins (Figure 1E). The correlations among second cousins varied somewhat between replicates (we will show below that we can control these correlations experimentally by applying molecular perturbations). If cellcycle length alone were inherited (e.g. by passing on regulators of the cell cycle to daughter cells), causing a correlation coefficient of $\rho}_{\mathrm{m}\mathrm{d}$ between mother and daughter cycle lengths, and sisters are correlated by $\rho}_{\mathrm{s}\mathrm{s}$, then first and second cousins would be expected to have cycle length correlation $\rho}_{\mathrm{s}\mathrm{s}}{\rho}_{\mathrm{m}\mathrm{d}}^{2$ and $\rho}_{\mathrm{s}\mathrm{s}}{\rho}_{\mathrm{m}\mathrm{d}}^{4$, respectively (Staudte et al., 1984). The actually observed cousin correlations are much larger, confirming previous observations on first cousins as summarized in Sandler et al. (2015) and extending them to second cousins. This discrepancy between simple theoretical expectation and experimental data was not due to spatial inhomogeneity or temporal drift in the data (Figure 1; Figure 1—figure supplement 3CE). Thus, the lineage trees show longranging intragenerational correlations that cannot be explained by the inheritance of cellcycle length.
Correlation patterns are explained by longrange memories of two antagonistic latent variables
We used these data to search for the minimal model of cellcycle control that accounts for the observed correlation pattern of lineage trees (Materials and methods and Appendix 2). To be unbiased, we assumed that cycle length $\tau $ is controlled jointly by a yet unknown number $d$ of cellular quantities that are inherited from mother to daughter, $\mathit{x}=({x}_{1},\dots ,{x}_{d}{)}^{\mathsf{T}}$, such that $\tau =\tau ({\sum}_{l=1}^{d}{\alpha}_{l}{x}_{l})$, with positive weights $\mathit{\alpha}$. We take $\mathit{x}$ to be a Gaussian latent variable and, generalizing previous work (Cowan and Staudte, 1986), describe its inheritance by a generic model accounting for intergenerational inheritance as follows: In any given cell $i$, $\mathit{x}}^{i$ is composed of an inherited component, determined by $\mathit{x}$ in the mother, and a cellintrinsic component that is uncorrelated with the mother. The inherited component is specified by an inheritance matrix $\mathbf{\mathbf{A}}$, such that the mean of $\mathit{x}}^{i$ conditioned on the mother’s $\mathit{x}$ is $\u27e8{\mathit{x}}^{i}\mathit{x}\u27e9=\mathbf{A}\mathit{x}$ (Figure 2A). The cellintrinsic component causes variations around this mean with covariance $\u27e8({\mathit{x}}^{i}\mathbf{A}\mathit{x})({\mathit{x}}^{i}\mathbf{A}\mathit{x}{)}^{\mathsf{T}}\mathit{x}\u27e9=\mathbf{I}$, where, with appropriate normalization of the latent variables, $\mathbf{\mathbf{I}}$ is the unit matrix. Additional positive correlations in sister cells may arise due to inherited factors accumulated during, but not affecting, the mother’s cycle (Arora et al., 2017; Barr et al., 2017; Yang et al., 2017); additional negative correlations may result from partitioning noise (Sung et al., 2013). These are captured by the crosscovariance between the intrinsic components in sister 1 and 2, $\u27e8({\mathit{x}}^{1}\mathbf{A}\mathit{x})({\mathit{x}}^{2}\mathbf{A}\mathit{x}{)}^{\mathsf{T}}\mathit{x}\u27e9=\gamma \mathbf{I}$. In total, $d(d+1)$ parameters can be adjusted to fit the correlation pattern of the lineage trees: the components ${a}_{lm}$ of the inheritance matrix $\mathbf{\mathbf{A}}$, the weights ${\alpha}_{l}$ and the sister correlation $\gamma $. Together, these inheritance rules specify bifurcating firstorder autoregressive (BAR) models for multiple latent variables governing cellcycle duration.
To determine the most parsimonious BAR model supported by the experimental data, we employed a standard Bayesian model selection scheme. Selection is based on the Bayesian evidence, which rewards fit quality while naturally penalizing models of higher complexity (defined as being able to fit more diverse data sets; for details see Appendix 2, Evidence calculation). Specifically, we evaluated the likelihood of the measured lineage trees for a given BAR model, used it to compute the Bayesian evidence, and ranked BAR models accordingly (Figure 2B).
The simplest model that generated high intragenerational correlations was based on the independent inheritance of two latent variables (Model III; Figure 2C, cyan dots), whereas onevariable models failed to meet this criterion (Model II, Figure 2C, blue dots and Model I). However, Model III consistently overestimated ancestral correlations and hence its relative evidence was low (<10% for all data sets). To allow additional degrees of freedom, we accounted for interactions of latent variables. The most general twovariable model with bidirectional interactions (Model VI), overfitted the experimental data and consequently had low evidence. The models best supported by the data had unidirectional coupling, such that ${x}_{2}$ in the mother negatively influenced ${x}_{1}$ inherited by the daughters, that is with ${a}_{12}<0$ and ${a}_{21}=0$ (Figure 2B, Models IV, V and VII). Among these, Model VII, with a single inheritance parameter ${a}_{12}$, is simplest, but was not compatible with experimental replicate rep1 as it could not generate secondcousin correlations (Figure 2B,C). Both Models IV and V were compatible with all replicates; however, Model V with only one selfinheritance parameter for both variables (${a}_{11}={a}_{22}>0)$ was preferred (Model V, Figure 2B,C, orange dots). Model V produced a remarkable inheritance pattern (Figure 2D): Individually, both latent variables had longranging memories, with ∼50% decay over 2–3 generations. However, the negative unidirectional coupling crosscorrelated the variables negatively along an ancestral line, resulting in cyclelength correlations that essentially vanished after one generation. Nevertheless, strong intragenerational correlations were reproduced by the model due to longrange memories of latent variables together with positive sistercell correlations ($\gamma >0$). We conclude that the coexistence of rapidly decaying ancestral correlations and extended intragenerational correlations can be explained by the inheritance of two latent variables, one of which inhibits the other.
Cell size and speed of cellcycle progression are antagonistic heritable variables
During symmetric cell division, both cell size and regulators of cellcycle progression are passed on from the mother to the daughter cells (Spencer et al., 2013; Yang et al., 2017; Arora et al., 2017; Barr et al., 2017). We now show that simple and generic inheritance rules for these two variables provide a physical realization for BAR Model V.
To divide, cells need to both grow to a minimum size (Shields et al., 1978) and receive license to progress through the cell cycle from the regulatory machinery (Novak et al., 2007). Indeed, growth and cellcycle progression can be separately manipulated experimentally in mammalian cells (Fingar et al., 2002). In particular, cells continue to grow in size when regulatory license is withheld, for example in the absence of mitogens, and growth is not otherwise constrained, for example by mechanical force or growth inhibitors (Fingar et al., 2002; Conlon and Raff, 2003).
While growth and cellcycle progression are separable and heritable processes, they also interact. At the very least, the length of the cell cycle needs to ensure that cells grow to a sufficient size for division. This interaction alone implies an effect of one inherited variable, cycle progression, on the other, cell growth, that anticorrelates subsequent cell cycles (as required by BAR model V): If a delayed regulatory license prolongs the mother’s cell cycle, it will grow large. By size inheritance, its daughters will be large at birth, reach a size sufficient for division quickly and hence may have shorter cell cycles. Thus, despite inheritance of growth and cellcycle regulators mothers and daughters may have very different cycle lengths due to this interaction.
Based on these ideas, we formulated a simple quantitative model of growth and cellcycle progression on cell lineage trees. We introduced the variables ‘cell size’ $s$, measuring metabolic, enzymatic and structural resources accumulated during growth, and $p$, characterizing the progression of the cellcycle regulatory machinery. Unlike the latent variables of the BAR model ${x}_{1}$ and ${x}_{2}$, their mechanistic counterparts $s$ and $p$, respectively, are governed by rules reflecting basic biological mechanisms (Figure 3A, Appendix 3). Size $s$ grows exponentially and is divided equally between the daughters upon division. We found that under some experimental conditions generating long cell cycles (downregulation of MYCN, see below), stable cell size distributions required feedback regulation of growth rate, as seen experimentally (Sung et al., 2013; Tzur et al., 2009); we implemented this as a logistic limitation of growth rate at large sizes for these conditions. The progression variable $p$ determines the time taken for the regulatory machinery to complete the cell cycle, which is controlled by the balance of activators and inhibitors of cyclindependent kinases. These regulators are inherited across mitosis (Spencer et al., 2013; Yang et al., 2017; Arora et al., 2017; Barr et al., 2017) and hence the value of $p$ is passed on to both daughter cells with some noise. Cells divide when they have exceeded a critical size, requiring time ${\tau}_{g}$, and the regulatory machinery has progressed through the cycle, which takes an approximately lognormally distributed time (Ryl et al., 2017; Mitchell et al., 2018) modeled as ${\tau}_{p}=\mathrm{exp}(p)$. Hence the cycle length is $\tau =\mathrm{max}({\tau}_{g},{\tau}_{p})$. Apart from requiring a minimum cell size for division, the growthprogression model does not implement a drive of the cell cycle by growth and thus allows cells to grow large during long cell cycles. By this mechanism, the cell size variable $s$ is influenced by cycle progression, analogous to the BAR variable ${x}_{1}$. By contrast, the progression variable $p$ is not influenced by cell size, analogous to the variable ${x}_{2}$ in the BAR model.
We fitted this model to the measured lineage trees by Approximate Bayesian Computation (Figure 3—figure supplement 1A and Figure 3—source data 1). The parameterized model yielded a stationary cell size distribution (Figure 3—figure supplement 1B) and reproduced the cyclelength distribution (Figure 3B and Figure 3—figure supplement 1C) as well as the ancestral and intragenerational correlations (Figure 3C and Figure 3—figure supplement 1D). Thus, the dynamics of cell growth and cellcycle progression, coupled only through a minimumsize requirement, account for the intricate cyclelength patterns in lineage trees.
To gain intuition on the inheritance patterns of cycle length, we first considered ancestral correlations, focusing on motherdaughter pairs. Individual cell cycles in the model are either growthlimited, that is division happens upon reaching the minimum size, or progressionlimited, that is the cell grows beyond the minimum size until the cycle is completed (Figure 3D and Figure 3—figure supplement 1E). If both mother and daughter are progressionlimited (i.e., the threshold size is exceeded by both), their cycles are positively correlated (Figure 3E, green dots). As in this case size inheritance is inconsequential, this positive correlation is explained by the inheritance of the cellcycle progression variable $p$ alone. By contrast, all motherdaughter pairs that involve at least one growth limitation show nearzero (Figure 3E, cyan dots) or negative correlations (Figure 3E, magenta and black dots). This pattern is explained by the anticorrelating effect that daughters of longerlived and hence larger mother cells require on average shorter times to reach the size threshold. Next, we considered intragenerational correlations, focusing on first cousins (Figure 3F). While cousins are positively correlated overall, this correlation is carried specifically by cousins that descend from a grandmother with a progressionlimited cell cycle (Figure 3F, blue dots), whereas cousins stemming from a growthlimited grandmother are hardly correlated (Figure 3F, orange dots). Since progressionlimited cells can grow large, this observation indicates that cousin correlations are mediated by inheritance of excess size, as is confirmed by conditioning cousin correlations on grandmother size (Figure 3—figure supplement 1F). Size inheritance over several generations is also evident in the autocorrelation of the time required to grow to minimum size, ${\tau}_{g}$ (Figure 3G and Figure 3—figure supplement 1G, black dots). The autocorrelation of the progression time ${\tau}_{p}$ is also positive (but less longranging; Figure 3G and Figure 3—figure supplement 1G, green squares), while the negative interaction with growth is reflected in the negative crosscorrelation (Figure 3G and Figure 3—figure supplement 1G, red triangles). In sum, the longrange memories of cellcycle progression and cell growth are masked by negative coupling of these processes, causing rapid decay of cellcycle length correlations along ancestral lines (Figure 3G and Figure 3—figure supplement 1G, orange triangles). These inheritance characteristics of the growthprogression model mirror those of BAR model V (see Figure 2D).
Effects of molecular perturbations on cellcycle correlations are correctly predicted by the model
If the growthprogression model captures the key determinants of the cellcycle patterns in lineage trees, it should be experimentally testable by separately perturbing growth versus cellcycle progression. We first derived model predictions for these experiments. Intuitively, if growth limitation were abolished by slowing cellcycle progression, only progression would be inherited and therefore motherdaughter correlations of cycle time could no longer be masked. As a result, we expect intragenerational correlations to be reduced relative to ancestral correlations when inhibiting cycle progression; conversely, inhibiting growth (and thereby increasing growth limitation) should raise them. Indeed, using the model to simulate perturbation experiments (Figure 3—source data 1), we found that growth inhibition increased cousin correlations relative to motherdaughter correlations, whereas slowing cellcycle progression decreased these correlations (Figure 4A).
To test this prediction, we slowed cellcycle progression experimentally by reducing MYCN, exploiting the doxycyclinetunable MYCN gene integrated in the TET21N cells. Cells grew to larger average size (Figure 4B, blue lines) over longer and more variable cell cycles (Figure 4C). These data show that lowering MYCN slowed cellcycle progression while allowing considerable cell growth. Further consistent with this phenotype, expression of mTOR, a central regulator of metabolism and growth (Fingar et al., 2002), was not lowered (Figure 4—figure supplement 1A). In a separate experiment, we inhibited cell growth by applying the mTOR inhibitor rapamycin, which reduced cell size by a small but reproducible amount (Figure 4B, red lines). This treatment also lengthened the cell cycle slightly (Figure 4C) but without changing MYCN protein levels (Figure 4—figure supplement 1B). Thus, lowering MYCN and inhibiting mTOR are orthogonal perturbations that act on cellcycle progression and cell growth, respectively. As predicted by the growthprogression model, these perturbations resulted in markedly different cyclelength correlation patterns within lineage trees (Figure 4D,E and Figure 4—figure supplement 1C,D): Lowering MYCN decreased intragenerational correlation and, in particular, removed secondcousin correlations. By contrast, rapamycin treatment strongly increased intragenerational correlations and caused ancestral correlations to decline only weakly. Collectively, these findings support the growthprogression model of cellcycle regulation.
Cousin correlations reflect active cellsize checkpoint
We then asked whether the cyclelength correlation patterns experimentally observed in lineage trees contain information about the underlying regulation. To this end, we fitted the BAR and growthprogression models to MYCN and rapamycin perturbation data. We again obtained good agreement with the data (Figure 4C,D and Figure 4—figure supplement 1C–E,G–J). MYCN knockdown cells grew larger; to obtain a stable cellsize distribution for the corresponding model fits, we implemented logistic regulation of growth rate at large cell sizes. When we applied, for comparison, this regulation to the control and rapamycin treatment data, the model fits (Figure 4—figure supplement 2) were not noticeably affected compared to purely exponential growth. This result indicates that growth rate regulation affecting large cells, as found experimentally (Ginzberg et al., 2018), is compatible with longrange intragenerational correlations in cellcycle length.
In terms of fit parameters of the model, MYCN inhibition caused considerable slowing of cycle progression and also a moderate decrease in growth rate (Figure 3—figure supplement 1A, parameters $\mu$ and $k$, respectively). As a result, the vast majority of cell cycles were progressionlimited (Figure 4F and Figure 4—figure supplement 1F). For the rapamycintreated cells, we estimated growth rates that were lower than for the control experiments on average, as expected (Figure 3—figure supplement 1A, parameter $k$). Also, correlations in cellcycle progression increased in motherdaughter and sister pairs (parameters $a$ and $\gamma $, respectively). This is consistent with rapamycin inhibiting mTOR and hence growth, but not affecting drivers of the cell cycle, ERK and PI3K (Adlung et al., 2017), since then lengthening of the cell cycle due to slower growth may allow prolonged degradation of cellcycle inhibitors, which would increase inheritance of cellcycle length (Smith and Martin, 1973). Taken together, rapamycin treatment increased the fraction of growthlimited cell cycles (Figure 4F and Figure 4—figure supplement 1F) and inheritance of cellcycle progression speed, thus causing increased ancestral and intragenerational correlations.
We hypothesized that growth may be limiting primarily for rapidly proliferating cell types, even without specific growth inhibition. We analyzed timelapse microscopy data of nontransformed mouse embryonic stem cells (Filipczyk et al., 2015) that proliferate much faster than the neuroblastoma cells (Figure 5A and Figure 5—figure supplement 1A). Sidebranch correlations of cycle length were again large (Figure 5B Figure 5—figure supplement 1B), as seen in the previous data except for the MYCNinhibited cells. Interestingly, the strength of the intragenerational correlations was most similar to the much more slowly dividing rapamycintreated cells (cf. Figure 4D). As before, the BAR model required two negatively coupled variables to account for these data (Figure 5C, Figure 5—figure supplement 1D,E). Fitting the growthprogression model to the data (Figure 5A,B), we found that the majority (∼60%) of cell cycles were limited by growth (Figure 5D, Figure 5—figure supplement 1C), indicating that cycle length of fast proliferating mammalian cells is, to a large extent, controlled by growth.
Discussion
Here, we showed that the seemingly paradoxical pattern of cellcycle lengths in lineage trees, with rapidly decaying ancestral and longrange intragenerational correlations, can be accounted for by the inheritance of two types of quantities: resources accumulated during the cell cycle (cell ‘size’) and regulators governing the speed of cellcycle progression. The fact that these are fundamental processes in dividing cells may help explain the ubiquity of the paradoxical cellcycle pattern. Targeted experimental perturbations of cell growth and cellcycle progression support our model.
As an alternative mechanism underlying the observed cellcycle variability and cousin correlations, modulation of the cell cycle by the circadian clock has been suggested, with strongest experimental evidence to date for cyanobacteria (Sandler et al., 2015; Mosheiff et al., 2018; Py et al., 2019). Unlike cyanobacteria, proliferating mammalian cells show entrainment of the circadian clock to the cell cycle with periods well below 24 hr (Bieler et al., 2014; Feillet et al., 2014). Whether in this setting the circadian clock could still influence cellcycle correlations remains to be studied.
Our proposed mechanism for cycle length correlations was motivated by Bayesian model selection. All data sets which displayed longrange correlation patterns (control replicate rep1 and rapamycin treatment of neuroblastoma cells as well as ESCs) selected a BAR model featuring longrange inheritance of two memory variables, which is masked in motherdaughter pairs by an anticorrelating interaction between them. The growthprogression model shares these essential features. Size inheritance is predicted to be particularly longrange by the fits of the growthprogression model to the experimental data. This memory emerges because growth does not force cellcycle progression, allowing cells to grow large in progressionlimited cycles and then pass on size over several generations. Such weak coupling between concurrent processes echoes recent work in E. coli on timing of cell division (Micali et al., 2018) and on the existence of two parallel adder processes for replication and division (Witz et al., 2019). Of note, the latter paper also utilizes a statistical framework for systematic model selection against experimentally observed correlation patterns akin to our approach.
Our findings raise the question of the mechanistic modes by which cells coordinate cell cycle and growth, which has been a longstanding problem; see Shields et al. (1978) for historical references and Ho et al. (2018) for a recent review. Recent work on this problem has shown the existence of negative feedback of cell size on growth rate (Ginzberg et al., 2018). We have found that our results remain robust when implementing a simple form of such a feedback (logistic dependence of growth rate on size) in the growthprogression model. Another recent study has shown that growth of many mammalian cells during the cell cycle adds a volume that only weakly increases with cell volume at birth (termed nearadder behavior, Cadart et al., 2018). This behavior appears to be caused by a combination of growthrate regulation and cellsize effects on cellcycle progression. We expect that factoring in the cellcycle length correlations studied here will help uncover the mechanistic details of cell size regulation. Refining our model in this direction may also help capture yet more detail of the correlation structure, such as the apparent increasing trend from aunt to first cousins and greataunt to second cousins. Moreover, we envisage that our inference approach could be extended to include finely resolved data on cellcycle phases (Chao et al., 2019) as well as multiple cell fates via asymmetric division and differentiation (Duffy et al., 2012).
Materials and methods
Experimental methods
Request a detailed protocolMYCNtunable mammalian neuroblastoma SHEP TET21N (RRID:CVCL_9812) cells were cultured as in Lutz et al. (1996). These TET21N cells were obtained from Dr. Frank Westermann (German Cancer Research Center, DKFZ) whose lab generated this line. This cell line is regularly authenticated by an inhouse DKFZ service using STR profiling. Mycoplasma contamination testing was negative. MYCN and mTOR were inhibited using 1 μg/ml doxycycline or 20–40 nM rapamycin (Calbiochem, 553210–100 UG), respectively. Cells were grown on ibidi μslides and phase contrast images (Nikon TiE) acquired every 6–15 min for up to 7 days under controlled growth conditions. The presented data consists of independent biological and technical replicates with n = 3 for untreated TET21N cells, n = 2 for MYCNinhibited and rapamycintreated cells. Cells were tracked in Fiji (version 1.48d) using the tracking plugin MTrackJ (Meijering et al., 2012). For flow cytometry, cells were stained with MYCN primary antibody (Santa Cruz Biotechnology, Cat# sc53993; RRID:AB_831602), secondary fluorescenceconjugated antibody goat antimouse Alexa Fluor 488 IgG (Life Technologies Cat#A11001; RRID:AB_2534069) and measured on a Miltenyi VYB MACSQuant Analyser. See Appendix 1 for details.
Data analysis
Request a detailed protocolMATLAB (R2016b) was used for all data analyses. Correlations represent Spearman rank correlations or, for the BAR model, Pearson correlation coefficients between the Gaussiantransformed cycle times. The difference between these two methods was far smaller than the experimental error. Error bounds were estimated by bootstrap resampling on the level of lineage trees. Censoring bias was avoided by truncating lineage trees after the last generation completed by all lineages within the experiment (see e.g. Sandler et al., 2015 and Appendix 1), truncating the trees to 7, 6 and 5 generations for the three MYCN amplified experiments. MYCNinhibited and rapamycintreated trees were five generations deep.
Bifurcating autoregressive (BAR) model
Request a detailed protocolWe constructed BAR models of cellcycle inheritance, as described in detail in Appendix 2. Briefly, the cell state is determined by a vector of Gaussian (latent) variables which are inherited from the mother to the daughter cells by a linear map plus a cellintrinsic noise term, which is correlated between daughters. The model is thus a Gaussian latentvariable model, where inheritance takes the form of an autoregressive vectorAR(1) process defined on a lineage tree. The cycle time is then calculated by an dataderived (approximately exponential) function of a weighted sum of the cellular state. We calculated wholelineage tree loglikelihood functions analytically and used them to evaluate Bayesian Evidences (Bayes factors) that quantify the relative support from the data for various model variants.
The growthprogression model
Request a detailed protocolCellcycle progression is modeled by a fluctuating, centered Gaussian heritable variable $q$, analogous to version II of the BAR model. Variables were scaled and shifted, $p={\sigma}_{p}q+\mu $, yielding lognormal progression durations ${\tau}_{p}=\mathrm{exp}(p)$. Size accumulation was modeled by exponential growth or for MYCN inhibition logistic growth. The normalized critical cell size ${s}_{\text{th}}$ fluctuates slightly and independently in each cell as ${s}_{\text{th}}=1+\zeta $ with $\zeta \sim \mathcal{N}(0,{\sigma}_{g}^{2})$. The growthprogression model was implemented in Matlab (R2016b), R (3.4.3) and OCaml (4.06) and 30 trees of 7 generations simulated, corresponding to the experimental dataset sizes. The simulation was repeated 100 times to generate confidence bounds. Parameters were fitted using Approximate Bayesian Computation independently for each dataset. See Appendix 3 for details.
Appendix 1
Experimental methods
Cell culturing and treatment
MYCNtunable mammalian neuroblastoma SHEP TET21N (TET21N, RRID: CVCL_9812) (Lutz et al., 1996) cells were cultured in RPMI 1640 medium supplemented with 10% fetal calf serum and 1% penicillin/streptomycin at 37°C 5% CO_{2} and 88% humidity. Versene was used for harvesting. TET21N cells were originally isolated from a female patient. Cell lines are authenticated by the German Cancer Research Center in house facility every halfyear. MYCNinhibited populations were established by incubating cells with 1 μg/ml doxycycline for 48–72 hr prior to further analysis. Growthinhibited populations were generated by treating cells with the mTOR inhibitor rapamycin (Calbiochem, 553210–100 UG) at 20 nM or 40 nM rapamycin dissolved in DMSO. Cells were treated with rapamycin or the same concentration of DMSO for 72 hr prior to harvesting for flow cytometry or livecell microscopy.
Livecell microscopy
10^{3} cells were grown on 8well ibidi μslides coated with collagen IV (Cat# 80822) in RPMI 1640 medium and imaged every 6–15 min for up to 7 days under controlled growth conditions at 37°C, 5% CO_{2} and 80% humidity (Pecon incubator P). Growth media was changed every 2–3 days. Phase contrast images were acquired with an inverted widefield microscope (Nikon TiE) using an EMCCD camera (Andor iXON3 885) and a 10x (CFI Planfluor DL10x, NA 0.3) or 20x lense (CFI Plan Apochromat DM 20x, NA 0.75). Cells were tracked in Fiji (version 1.48d) using the manual tracking plugin MTrackJ (Meijering et al., 2012). The presented imaging data consists of independent biological and technical replicates with n = 3 for untreated TET21N cells, n = 2 for MYCNinhibited cells and n = 2 for rapamycintreated cells.
Flow cytometry and antibody staining
10^{6} cells were fixed with 4% paraformaldehyde for 15 min at room temperature. Permeabilization was performed in 90% icecold methanol for at least 24 hr at −20°C. Cells were washed in staining buffer (1% BSA, 0.1% TritonX in PBS), and incubated with 0.5–1 μg per sample of MYCN primary antibody (Santa Cruz Biotechnology, Cat# sc53993; RRID:AB_831602) for 1 hr at room temperature. Cells were washed 3x with staining buffer and incubated with a secondary flouresenceconjugated antibody, goat antimouse Alexa Fluor 488 IgG (Life Technologies Cat#A11001; RRID:AB_2534069), again for 1 hr at room temperature. Cells were washed 3x with staining buffer and DNA content staining was performed with FxCycle Violet Stain (Thermo Fischer Scientific). A Miltenyi VYB MACSQuant Analyser was used for measurements and data was analysed using FlowJo software.
Transcriptomics
mTOR mRNA expression data in TET21N cells under control and MYCNinhibited conditions was obtained from RNASeq measurements by Ryl et al. (2017) deposited at GEO (GSE98274).
Data analysis
MATLAB (R2016b) was used for all data analysis steps.
Correlation coefficients
Correlation patterns of lineage trees of data and the growthprogression model represent Spearman rank correlations. For the bifurcating autoregressive (BAR) model, Pearson correlation coefficients are calculated between the Gaussiantransformed cycle times. These are the same as Gaussian rank correlations between cycle times. In practice, the differences between Gaussian rank correlations and Spearman rank correlations were much smaller than experimental error bounds on either, so we did not differentiate between the measures. Confidence bounds on correlation coefficients were estimated by bootstrap resampling on the level of lineage trees; resampling on the level of individual pairs of related cells would neglect the correlations between cells of the same tree and thus underestimate variability. Trees that did not contain information on the cell pair under analysis were removed prior to bootstrapping. At each bootstrap repeat, family trees were randomly drawn with replacement up to the number of trees in the original dataset. From the resulting bootstrap sample all cell pairs were used to calculate the sample correlation coefficient. This process was repeated 10,000 times. From the resulting distribution of correlation coefficients the 95% quantiles were used as confidence bounds.
Exponential growth
For each experiment, an exponential growth model of the form ${N}_{t}={N}_{0}\mathrm{exp}(kt)$ was fitted to cell counts over time by performing a maximum likelihood estimation using the trustregion algorithm in MATLAB.
Cellcycle length distribution over time
At each time point, the movingwindow median was calculated from all cells born within a window of 10 hr before or after this point.
Randomization
All cells within a dataset were randomly paired with each other and correlation of the resulting sample calculated. This procedure was repeated 10,000 times. From the resulting distribution the mean and the 95% quantiles are given.
Censoring
Censoring bias resulting from a finite observation time can lead to an overrepresentation of faster cells. To demonstrate this effect, we generated trees using a toy model with independent normally distributed cycle lengths. The trees were truncated at various total observation times and Spearman rank correlation coefficients between all related cells within the observation time window were sampled. As Figure 1—figure supplement 3A shows, short observation times strongly distort the sampled motherdaughter and sibling correlations away from their true value 0. The same basic effect persists for more distant relationships and can be further enhanced if cycle lengths are inherited. This censoring bias can be avoided by truncating lineage trees not after a given observation time, but after the last generation completed by all lineages within the experiment, uniformly over all trees (see e.g. Sandler et al., 2015). In this way slower and fastercycling lineages are represented equally. Because some cells were inevitably lost from the field of view by migration, and a small percentage of cells showed extremely slow cycles, such a strict cutoff was unfeasible in our experiments. We assumed that cell loss by migration is not correlated to cycle length, so migrating cells were not counted as missing from otherwise complete trees. Within the remaining tree, we then determined the last generation to be included in our analysis by the following procedure: We first counted the number of cells ${n}_{\text{alive}}(G)$ within each generation $G$ that were still alive at the end of the observation period. The last generation ${G}_{\text{last}}$ to be included was then determined as the maximum generation such that
All further generations were removed from the dataset. This procedure truncated the trees to 7, 6 and 5 generations, respectively for the three replicate experiments using our MYCN amplified cell line. MYCNinhibited and rapamycintreated trees were 5 generations deep.
Spatial trend
To assess potential spatial biases related to locally variable conditions, cells were divided into a 4 × 4 grid according to their position at division. The cycle length distribution of cells within each grid region (containing ≥ 5 cells) was compared to the distribution (1) within every other grid region and (2) of the whole dataset at a 5% significance level using a twosided KolmogorovSmirnov test and correcting for multiple testing using the BenjaminiHochberg procedure (using functions ks.test and p.adjust in R). Note also that because cells are motile they experience a range of local environments during their lifetime.
Appendix 2
Bifurcating autoregressive model for cycle inheritance
Setup
In order to explore systematically which simple local inheritance schemes can generate the experimentally observed cycle length correlations, we study a class of Gaussian latent variable models of adjustable complexity. In the models, the cycle length $\tau $ of a cell is obtained from a standard normal variable $\mathrm{\ell}$ by a nonlinear transformation $\tau =g(\mathrm{\ell})$. In our data, cell cycle lengths are roughly lognormally distributed, so $g$ is approximately a shifted exponential function. To simplify and make the model more robust to outliers, we determine $g$ empirically, such that its inverse ${g}^{1}$ transforms the cell cycle length into a standard normal variate. That is, we choose $g(\mathrm{\ell})={c}_{\text{ex}}^{1}({c}_{\text{gauss}}(\mathrm{\ell}))$ where ${c}_{\text{gauss}}$ is the cumulative distribution function (CDF) of a standard normal distribution, and ${c}_{\text{ex}}$ is the empirical CDF of the experimental data. This transformation discards all information about the shape and mean value of the cycle length distribution; the set of $\mathrm{\ell}$ variables then purely reflect the strength of correlations between cell cycles. The (Pearson) moment correlation coefficients between the variables $\mathrm{\ell}$ are identical to the socalled Gaussian rank correlation coefficients (Boudt et al., 2010) between the corresponding cycle lengths $\tau $, which are similarly robust to outliers as the more common Spearman rank correlation.
The Gaussian variable $\mathrm{\ell}$ is used to model correlation by inheritance. $\mathrm{\ell}$ is a weighted sum of $d$ latent, centered Gaussian variables $\mathit{x}=({x}_{1},\dots ,{x}_{m}{)}^{\mathsf{T}}$ with positive weights $\mathit{\alpha}=({\alpha}_{1},\dots {\alpha}_{m}{)}^{\mathsf{T}}$, denoted as vectors $\mathit{x}$ and $\mathit{\alpha}$. That is, $\ell ={\mathit{\alpha}}^{\mathsf{T}}\mathit{x}=\sum _{l}{\alpha}_{l}{x}_{l}$. Inheritance in the model occurs by passing on latent variables from mother to daughter cells. The basic model equation relation reads
Here, a superscript $i=1,2$ denotes a daughter cell, and absence of a superscript refers to the mother cell. The matrix $\mathbf{\mathbf{A}}$ implements inheritance: The average of a daughter’s latent variables, given the mother’s is $\u27e8{\mathit{x}}^{i}\mathit{x}\u27e9=\mathbf{A}\mathit{x}$. This linear coupling of latent variables through inheritance may take any form compatible with the basic stability requirement that its operator norm must satisfy $\parallel \mathbf{\mathbf{A}}\parallel <1$. Since both daughters inherit the same contribution from the mother, inheritance correlates the daughters’ latent variables positively. Daughter cells are also subject to random fluctuations which we model by standard normal random vectors $\mathit{\xi}}^{i$. These fluctuations are correlated due to the term $\overline{b}{\mathit{\xi}}^{\overline{\u0131}}$ in Equation 2. Here $\overline{\u0131}$ designates the sister cell of $i$, for example $\overline{2}=1$. We parametrize these correlations via
The sister correlations conditioned on the mother latent variables then become
where $\mathbf{\mathbf{I}}$ is the $d$dimensional unit matrix. Positive sister correlations $(\gamma >0)$ may arise due to fluctuations that occur within the mother cell after its cycle duration has been fixed and are shared by the daughters. Negative correlations $(\gamma <0)$ may arise due to partitioning noise upon inheritance. Note that latent variable fluctuations are correlated between sisters but uncorrelated between different latent variables. Effectively, our choice of parametrization partitions all fluctuating cell cyclerelevant processes within the daughter cells into $d$ Gaussian components that are maximally decorrelated, similar to a principal component decomposition.
Overall, Equation 2 defines an unbiased model with linear, local inheritance of latent variables, and an output that is a linear combination of latent variables. Its Gaussian form may be justified as the maximumentropy distribution (Jaynes, 1957) for this problem, since only covariance information is used as an experimental input at this stage. Our model is a firstorder vector autoregressive process, defined on the lineage tree of cells; it is a latentvariable generalization of the bifurcating autoregressive model already considered by Staudte and coworkers (Staudte et al., 1984). We remark that in Staudte et al. (1984) the dimensionality is $d=1$, and therefore $\mathit{\alpha}$ is unnecessary.
Stationary exponential growth
Combining Equations 2 and 4 by the law of total variance, the latent covariance satisfies
In stationary exponential growth, averaging over a single lineage forward in time, a stationary distribution with mean $\mathbf{0}$ and covariance ${\mathbf{\mathbf{C}}}_{\mathrm{\infty}}$ is established. Then $\u27e8{\mathit{x}}^{i}{{\mathit{x}}^{i}}^{\mathsf{T}}\u27e9=\u27e8\mathit{x}{\mathit{x}}^{\mathsf{T}}\u27e9={\mathbf{C}}_{\mathrm{\infty}}$, and Equation 5 implies
We take this stationary distribution of latent variables as initial condition for root cells of lineage trees, assuming they come from an equilibrated growth phase. This assignment is not strictly correct because the stationary distribution along forward lineages is different from the distribution of all cells in an exponentially growing population at a given time (Lin and Amir, 2017); however, the difference was small for our parameters when tested numerically (not shown).
Computation of the likelihood
We aim to compute the probability of generating a lineage tree with given cycle lengths within the model. We fix a minimum generation number and consider only trees in which essentially all branches reach this number, thus discarding overhanging cells on some branches, (see also Sandler et al., 2015). This is crucial since in experiments with finite duration, selection bias would otherwise be introduced (Cowan and Staudte, 1986).
We begin by indexing cells in a tree by their pedigrees, which are the sequences of sister indices counting from the root cell, for example $I={i}_{1}{i}_{2}\mathrm{\dots}{i}_{k}$ for a cell in generation $k$ and $I{i}_{k+1}$ for one of its daughters. Sorting these indices, we can then arrange all Gaussiantransformed cycle lengths in a tree into a single vector $\underset{\mathbf{\_}}{\mathit{\ell}}$. Since $\underset{\mathbf{\_}}{\mathit{\ell}}$ is Gaussian with mean $\underset{\mathbf{\_}}{\mathbf{0}}$, its logprobability takes on the simple quadratic form
To evaluate Equation 7, we need to determine the joint covariance matrix ${\underset{\xaf}{\mathbf{\mathbf{C}}}}_{\mathrm{\ell}}$ of Gaussian cycle lengths over the given tree structure as a function of the parameters $\mathbf{A},\gamma ,\mathit{\alpha}$. We start by first deriving the joint covariance matrix $\underset{\xaf}{\mathbf{\mathbf{C}}}$ of the latent variables $\underset{\mathbf{\_}}{\mathit{x}}$. This is a block matrix with $d\times d$ blocks ${\mathbf{\mathbf{C}}}^{IJ}$ that correspond to pairs of cells in welldefined relationships, such as motherdaughter, cousincousin, etc. Since the lineage tree is sampled from stationary growth, ${\mathbf{\mathbf{C}}}^{IJ}$ depends only on the relationship of $I$ and $J$, that is on their respective ancestral lines up to the latest common ancestor, and not on the history before. In particular, if $I=J{i}_{k+1}\mathrm{\dots}$ then cell $I$ is a descendant of cell $J$ and we write this as $I>J$; otherwise we write $I\ngtr J$. Note that $I\ngtr I$.
Splitting one cell pedigree as $I=Ki$, from Equation 2 we derive the relations
Equation 8 lets us compute ${\mathbf{\mathbf{C}}}^{IJ}$ by a recursive procedure, as follows:
Consider the case $I>J$. If $I=Ki$, then $J\ngtr K$. Now use Equation 8 (i) repeatedly ($k$ times), moving up the ancestral line, until arriving at the form $\mathbf{C}}^{IJ}={\mathbf{A}}^{k}\u27e8{\mathit{x}}^{J}{{\mathit{x}}^{J}}^{\mathsf{T}}\u27e9={\mathbf{A}}^{k}{\mathbf{C}}_{\mathrm{\infty}$.
In the case $I=K{i}_{1}\mathrm{\dots}{i}_{k},J=K{j}_{1}\mathrm{\dots}{j}_{{k}^{\prime}}$ with ${j}_{1}={\overline{\u0131}}_{1}$, no cell is a descendant of the other, and their last common ancestor is $K$. Use Equation 8 (i) (or its transpose) on both branches repeatedly until the form $\mathbf{A}}^{k1}\u27e8{\mathit{x}}^{K{i}_{1}}{{\mathit{x}}^{K{\overline{\u0131}}_{1}}}^{\mathsf{T}}\u27e9{{\mathbf{A}}^{{k}^{\mathrm{\prime}}1}}^{\mathsf{T}$ is obtained. Then use Equation 8 (ii) to get ${\mathbf{\mathbf{C}}}^{IJ}={\mathbf{\mathbf{A}}}^{k}{\mathbf{\mathbf{C}}}_{\mathrm{\infty}}\mathbf{\mathbf{A}}^{{k}^{\prime}}{}^{\U0001d5b3}+\gamma {\mathbf{\mathbf{A}}}^{k1}\mathbf{\mathbf{A}}^{{k}^{\prime}1}{}^{\U0001d5b3}$
These two cases cover all possible cellcell relations, so that the procedure fully determines the joint latent covariance $\underset{\xaf}{\mathbf{\mathbf{C}}}$ for a given tree structure, inheritance matrix $\mathbf{\mathbf{A}}$ and sister correlation $\gamma $.
Finally, to obtain the covariance ${\underset{\xaf}{\mathbf{\mathbf{C}}}}_{\mathrm{\ell}}$ of the Gaussian cycle lengths $\mathrm{\ell}$, we project onto $\mathit{\alpha}$. The elements of ${\underset{\xaf}{\mathbf{\mathbf{C}}}}_{\mathrm{\ell}}$ result as
This completes the evaluation of the logprobability $\mathcal{P}$ (Equation 7), which is also equal to the loglikelihood of the model, $\mathcal{\wp}(\underset{\mathbf{\_}}{\mathit{\ell}})=\mathcal{\mathcal{L}}(\mathbf{A},\gamma ,\mathit{\alpha})$. Accounting for the constraint $\u27e8{\mathrm{\ell}}^{2}\u27e9=1$ which we impose to fix the arbitrary normalization of $\mathrm{\ell}$, the full model has ${d}^{2}+1+d1=d(d+1)$ adjustable parameters. This number can be reduced by restricting the inheritance matrix to a specific form, or by setting $\gamma =0$, as was done for the model variants discussed in the main text.
As a corollary, the Gaussian rank correlation between cycle lengths of any pair of cells results as
In the onedimensional special case $d=1$, the projection on $\alpha $ becomes irrelevant and Equation 10 reduces to
where $k,{k}^{\prime}\ge 1$ the distances to the latest common ancestor as in the algorithm above, and $a\equiv \mathbf{\mathbf{A}}$ is the $1\times 1$ inheritance matrix. Some special cases of Equation 11 given already in Cowan and Staudte (1986) are ${\rho}_{\text{ss}}^{\text{gauss}}=[{a}^{2}+\gamma (1{a}^{2})]$ for sisters and $\rho}_{\mathrm{c}1}^{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}}={a}^{2}{\rho}_{\mathrm{s}\mathrm{s}}^{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}}={{\rho}_{\mathrm{m}\mathrm{d}}^{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}}}^{2}{\rho}_{\mathrm{s}\mathrm{s}}^{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}$ for first cousins. In other words, to compute the correlation between related cells, one multiplies motherdaughter correlations along the path connecting them, taking a shortcut via the daughters of the last common ancestor where one instead multiplies with the sistersister correlation. Specializing further to $\gamma =0$, Equation 11 reduces to the wellknown relation $\rho}_{{\tau}^{I}{\tau}^{J}}^{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}}={{\rho}_{\mathrm{m}\mathrm{d}}^{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}}}^{k+{k}^{\mathrm{\prime}}$ where $k+{k}^{\prime}$ is the number of cell divisions linking $I$ and $J$. As detailed in the main text, these onedimensional special cases are insufficient to explain our data.
Evidence calculation
To compare different model versions’ ability to explain but not overfit the data, we employed a standard Bayesian model selection scheme (see e.g. Wasserman, 2000; MacKay, 2003). Within this framework, model selection is treated on the same grounds as parameter inference; the task is to assign to each one out of a set $\mathcal{M}$ of models its likelihood to have generated the data. One or several plausible models can then be selected on these grounds. Concretely, the scheme proceeds as follows. The probability of model $M$ to generate data $\underset{\mathbf{\_}}{\mathit{\ell}}$ is obtained by integrating over all parameter values ${\pi}_{M}$, which are distributed over a parameter space ${\mathrm{\Pi}}_{M}$ with prior distribution $p({\pi}_{M}M)$:
Here, $p(\underset{\mathbf{\_}}{\mathit{\ell}}{\pi}_{M},M)=\mathrm{exp}[\mathcal{\wp}(\underset{\mathbf{\_}}{\mathit{\ell}})]=\mathrm{exp}[\mathcal{\mathcal{L}}({\pi}_{M})]$ is equal to the likelihood function for model $M$, Equation 7. Applying Bayes’ rule, the probability that among $\mathcal{M}$, $M$ was the model that generated the data, is then
When models are equivalent a priori as we assume here, then both the prior belief in a model, $p(M)$, and the entire denominator in Equation 13 are unimportant constants. Then the socalled evidence or Bayes factor, obtained by calculating
is proportional to each model’s probability of having generated the data, Equation 13. We calculated $E(M)$ numerically by MonteCarlo integration of Equation 14; in the main text we show the evidences relative to Model V. Conventionally, an advantage in $E$ of a factor of 10 or more is considered strong support in favor of a model.
We briefly discuss some important features of model selection by evidence. In the asymptotic case of large samples (not applicable for the present data), the evidence $E$ is approximated by the wellknown Bayes information criterion (BIC), which is an alternative to the popular Akaike information criterion (AIC). While AIC is constructed to select a model whose predictions are maximally similar to future repetitions of the same experiment, evidence and BIC select the model that is most likely to have generated the existing data. BIC and evidence, but not AIC, have a desirable consistency property: If the models $\mathcal{M}$ are recruited from a hierarchy of nested models which also contains the true model, then the simplest model in $\mathcal{M}$ comprising the true model is always favored for large enough samples (Wasserman, 2000). This consistency is a manifestation of a general preference of the evidence for parsimonious models. To illustrate this point, following MacKay (2003), we expand the log evidence around the maximum aposteriori estimate ${\pi}_{M}^{*}$, using Laplace’s method:
Here, ${H}_{M*}={\left(\frac{{\partial}^{2}\mathcal{L}}{{\partial}_{{\pi}_{M}^{i}}{\partial}_{{\pi}_{M}^{j}}}\right)}_{{\pi}_{M}^{*}}$ is the Hessian of the loglikelihood, and for simplicity we have assumed a flat parameter prior $p({\pi}_{M})=1/\mathrm{vol}({\mathrm{\Pi}}_{M})$. The ith eigenvalue $2\pi /{s}_{i}$ of ${H}_{M*}$ determines the width $s}_{i$ of the peak of the posterior distribution around ${\pi}_{M}^{*}$, along the ith principal axis. In the last equality, we have written the parameter space volume $\mathrm{vol}({\mathrm{\Pi}}_{M})={\prod}_{i}{S}_{i}$ as a product of parameter ranges $S}_{i$ along the principal axes. Equation 15 can be interpreted as follows: As more parameters are added to a model, the fit accuracy, measured by $\mathcal{L}({\pi}_{M}^{*})$, generally improves. However, each new parameter ${i}^{\prime}$ incurs a penalty $\mathrm{log}({s}_{{i}^{\mathrm{\prime}}}/{S}_{{i}^{\mathrm{\prime}}})<0$. The more the new parameter needs to be constrained by the data, the more the evidence is reduced. Thus the basic mechanism of parsimony in Bayesian model selection is this: Complex models are characterized by a large number of parameters with wide a priori allowed ranges and sensitive dependence on the data; in other words, they require the data to pick parameters from a large set of possibilities. Complex models are penalized and ranked as less likely. Indeed, such an overly flexible model can be fitted to diverse data, which we should expect to diminish the support that a particular set of data can give to it.
Numerical efficiency and implementation
The evaluation of $\mathcal{L}$ (Equation 7) requires a final numerical inversion of the recursively assembled covariance ${\underset{\xaf}{\mathbf{\mathbf{C}}}}_{\mathrm{\ell}}$ to obtain the stiffness matrix, which costs $O({N}^{3})$ operations, where $N$ is the number of cells in the tree. To reduce this cost, it is possible to devise an equivalent recursive scheme in which the projected stiffness matrix, not the covariance matrix, is computed recursively, using efficient blockinverse formulæin each recursive step. The computational complexity can then be improved to $O({N}^{2})$. A version of this alternative, equivalent scheme was implemented in the programming language OCaml and used to obtain the results presented in the main text.
Appendix 3
The growthprogression model
Setup
The growthprogression model is based on the idea that the two cellcycle controlling processes are cellcycle progression and cell growth. The state variables of both processes, namely, the timing of the regulatory license to divide encoded by $p$, and cell size $s$, respectively, are inherited from mother to daughter. The two processes are coupled via inheritance as detailed in the following.
The cellcycle progression process
Inheritance of the velocity of cellcycle progression is modeled by a fluctuating, centered Gaussian variable $q$, passed on from mother to daughter, entirely analogous to version II of the BAR model (see there for additional explanation), according to
where subscript $i=1,2$ denotes a daughter cell, $\overline{\u0131}$ its sister, and no subscript, the mother. $a$ with $a<1$, implements inheritance. The intrinsic fluctuation strength and coupling is given by $b$ and $\overline{b}$; effectively, daughter cells are correlated for given mother by, $\u27e8{q}_{i}{q}_{\overline{\u0131}}q\u27e9=\gamma ;\u27e8{q}_{i}^{2}q\u27e9=1,$ where $\gamma =2b\overline{b}$. From the centered $q$ variables, shifted and scaled Gaussian variables $p={\sigma}_{p}q+\mu $ were generated, finally yielding lognormal regulation cycle durations ${\tau}_{p}=\mathrm{exp}(p)$. The duration ${\tau}_{p}$ is the time elapsed since the last division until regulatory license is given to divide again. Overall, the progression process has four adjustable parameters, $\mu$, ${\sigma}_{p}$, $a$ and $\gamma $.
The growth process
The growth duration ${\tau}_{g}$ is defined as the time to grow from an initial size ${s}_{b}$ to the threshold size ${s}_{\text{th}}$. Size accumulation was modeled by exponential growth with the exponential growth rate
However, under MYCN inhibition, exponential growth was prone to generate unreasonably large cells. Here we instead modeled growth by the logistic growth process
where $k$ is the growth rate constant and ${s}_{\text{max}}$ the maximum cell size. We fixed ${s}_{\text{max}}=20$, to match the approximate cell size at which the growth rate starts to decrease with observations (Sung et al., 2013; Tzur et al., 2009). The two growth laws Equations 17, 18 yield
and
respectively.
The normalized threshold cell size ${s}_{\text{th}}$ fluctuates slightly and independently in each cell as ${s}_{\text{th}}=1+\zeta $ with $\zeta \sim \mathcal{N}(0,{\sigma}_{g}^{2})$. At division, the final mother cell size ${s}_{\text{div}}$ is halved, with each daughter receiving a new size at birth ${s}_{b}={s}_{\text{div}}/2$. Effectively, for the subset of cell cycles that are limited by growth, this process corresponds to a sizer mechanism Facchetti et al. (2017). The growth process has two adjustable parameters, $k$ and ${\sigma}_{g}$.
Coupling progression and growth
The two processes are coupled via a checkpoint which requires both to be completed before a cell has license to divide. The cellcycle length is then determined as $\tau =\mathrm{max}({\tau}_{p},{\tau}_{g})$. If division is stalled by insufficient cycle progression, $s$ continues to accumulate until cell division, so that the final cell size ${s}_{\text{div}}>{s}_{\text{th}}$. Thus, inheritance of the growth process is influenced by the cycle progression process. In contrast, the progression process is inherited in an autonomous fashion. This unidirectional inheritance structure recapitulates the unidirectional coupling between hidden processes found in the preferred BAR models IV and V.
Model simulation
The growthprogression model was implemented in Matlab (R2016b), R (3.4.3) and OCaml (4.06) (with identical results but increasing execution speed) and lineage trees were simulated. For each tree, an initial cell was generated with birth size ${s}_{b}={s}_{\text{th}}/2$ and $p=\mu $ and subsequently, an unbranched single lineage was simulated for 100 generations for equilibration. Its final cell was used as founder cell for the tree. For the data shown, 30 trees of 7 generations each were simulated, roughly corresponding to the dataset sizes obtained experimentally. The simulation was repeated 100 times to generate confidence bounds.
Parameter optimization
Parameters were fitted using Approximate Bayesian Computation independently for each dataset. In an adaptive procedure, (sometimes nonuniform) prior distributions of model parameters were first generated. For each parameter set, 500 trees were simulated at a depth of 7 generations. To compare data $D$ and simulations $\widehat{D}$, we used a set $S$ of summary statistics composed of the nine correlation coefficients as shown in Figure 3C, and the mean and all quartiles of the distribution of cellcycle durations. A squareddistance between data and simulation summary statistics was calculated as
where ${\sigma}_{i}$ were calculated based on the data bootstrap confidence (95%) bounds. Samples of the approximate posterior probability distribution of parameter values (Figure 3–figure supplement 3A ) were then generated by accepting parameter sets with ${\rho}^{2}(S(\widehat{D}),S(D))<\u03f5$, weighted by the inverse of the local prior density in parameter space. We used a tolerance $\u03f5=2$ but results were not sensitive to the choice of $\u03f5$ within the range $1\dots 4$. Prior parameter ranges were extended as far as necessary for accepted sets to converge. The final accepted ranges of each parameter was used as credible regions, and the medians of each parameter distribution from these samples were selected as bestfits for further analysis.
Data availability
Data generated or analysed during this study are included in the manuscript and supporting files. Source data files have been provided for Figures 1 and 4.

NCBI Gene Expression OmnibusID GSE98274. RNASeq of SHEP TET21N cells upon Doxorubicin treatment.
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Article and author information
Author details
Funding
Bundesministerium für Bildung und Forschung (0316076A)
 Thomas Höfer
Bundesministerium für Bildung und Forschung (01ZX1307)
 Thomas Höfer
Bundesministerium für Bildung und Forschung (031L0087A)
 Thomas Höfer
Seventh Framework Programme (FP7HEALTH2010 ASSET 259348)
 Thomas Höfer
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank Frank Westermann and Tatjana Ryl for TET21N cells and laboratory setup; Timm Schroeder, Fabian Theis and Carsten Marr for embryonic stem cell data and discussions; Alessandro Greco for differential gene expression analysis and all members of the Höfer group for discussions. Grant support to TH from BMBF (MYCNET 0316076A and SysmedNB 01Z × 1307), EU (FP7HEALTH2010 ASSET 259348), BMBF and EU (EraCoSysMed OPTIMIZENB 031L0087A), as well as DKFZ core funding are gratefully acknowledged; TH is a member of CellNetworks.
Version history
 Received: August 10, 2019
 Accepted: January 22, 2020
 Accepted Manuscript published: January 23, 2020 (version 1)
 Version of Record published: February 13, 2020 (version 2)
Copyright
© 2020, Kuchen et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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