Empirical lineage size distribution

In the experiment conducted by Esk et al. [5], cerebral organoids were grown from roughly 24,000 stem cells, genetically identical except for a distinct genetic barcode in each cell serving as a lineage identifier (LID). To determine the contribution of each initial stem cell to organoids of different ages, organoids were subjected to amplicon high-throughput sequencing. The sequencing reads (after filtering and error-correction) corresponding to each LID were counted, and the per-LID read counts were normalized to an approximate number of cells comprising each lineage (see Methods for details).

The resulting lineage size distribution (Fig 1A) shows, as expected, small and equally sized lineages for organoids harvested at day 1 (lineages sizes around 1 cell). The distribution grows more dispersed until day 11 (up to 30 cells/lineage) and extends over 4 to 5 orders of magnitude (up to 100,000 cells/lineage) after 40 days.

A common mathematical model for distributions extending over multiple orders of magnitude are so-called (Pareto) power laws. In power-law distributions, the frequency of objects of size l or larger is proportional to l^−α, where the parameter α controls the amount of dispersion. The parameter α is called (Pareto) equality index because a small value of α indicates a large dispersion (i.e. diverse sizes), while a large values of α indicates clustered (i.e. similar) sizes. Compared to other commonly used distributions such as the normal distribution, power-law distributions typically represent very dispersed data. In double-logarithmic frequency vs. size plots, power laws appear as straight lines with slope α. This fits the lineage size distribution on day 40 well for α≈0.46 (Fig 1B). We remark that α≈0.46 represents a small equality index (i.e. very dispersed lineage sizes); in applications of Pareto distributions values of α often lie between 1 and 2 [8].

At early time points, technical/experimental noise dominates over actual variations in linage size, and we consequently find large sample-to-sample variations of the equality index (Fig 1C) between the three replicates for each time point. Starting with day 11, all samples show an equality index close to α≈0.46 (Figs 1C and S1). During the same time, however, the dispersion of the linage size distribution grows considerably (Fig 1A).

Truncated Zipfian rank-size distribution

To understand why the equality index converges while the inequality of lineage sizes grows, we rank lineages by size (largest lineage first) and plot the resulting rank-size distribution in a double-logarithmic plot (Fig 1D). In such a plot, lineage sizes governed by a Pareto law with index α would be expected to be governed by a Zipfian power law (i.e. lineage sizes decrease proportional to r^−1/α with increasing rank r). Because of the double-logarithmic nature of the plot, the rank-size distribution would thus be expected to form a straight line with slope −1/α [9].

We observe that even from days 11 onwards (where we found α≈0.46), lineage sizes adhere to the Zipfian law only up to a certain lineage size threshold l_Th. Lineages larger than the threshold l_Th are multiple orders of magnitudes smaller than the Zipfian law would predict, and more uniform in size (Figs 1D and S1). For most time points (very early ones excluded), the majority of lineages lie below the threshold and thus in the Zipfian regime. The threshold l_Th itself grows rapidly with time, increasing more than 1,000-fold (Fig 1E) over 40 days. At the same time the spread of lineage sizes above the threshold l_Th (i.e. ratio between l_Th and largest lineage size) grows only by a factor of about two (from 8.5 to 21; Fig 1D). We conclude that lineages outside the Zipfian regime grow quickly (since the threshold grows rapidly) and roughly similarly fast (since the spread of linage sizes in this regime increases only slowly). Lineages in the Zipfian regime, in contrast, show no overall shift towards larger lineages sizes over time, indicating that growth has mostly ceased for these lineages.

Lineages switch growth regime

The threshold l_Th thus partitions lineages according to their growth regime into fast-growing and slow/non-growing. Of the (on average) 10,851 lineages that contribute to the final organoid 8,389 lineages fall into the fast-growing category on day 1; but on day 11 their number has dropped to 1,496, and on day 40 only 191 (about 2%) fast-growing lineages remain (Fig 1F). Lineages thus start in a fast-growing regime, and one by one switch to a regime of slow/no growth as time progresses. The later that switch occurs for a particular lineage, the bigger it has become before its growth ceases, leading to larger and larger lineages in the slow-growing regime and consequently to l_Th increasing as time progresses. Within the fast-growing regime (i.e. lineages sizes above l_Th), the difference in size between the largest and smallest lineage is about 1–2 orders of magnitude (Fig 1D). Within the slow-growing regime, the spread between lineage sizes reaches 3.5 orders of magnitude on day 40 (Fig 1D). We thus neglect the differences between lineages sizes within fast-growing lineages to arrive at a simplified representation of this model. In this cartoon version of our data-derived model, at any particular point in time all fast-growing lineages have the same size. When they drop out of the fast-growing regime (at a random point in time) their size arrests (Fig 1G).

This proposed lineage-specific switching from fast to slow growth can also quantitatively reproduce the observed truncated Zipfian with α = 0.046. One mathematically simple model are fast-growing lineages which grow exponentially with rate γ and drop out of the fast-growing regime with rate σ = αγ (S1 Supplemental Methods). But while this simple example assumes an unspecified biological mechanism behind the lineage-specific growth regime switches, we show in the following that no such mechanism is in fact necessary. Instead, we show that such growth regime switches emerge naturally from a cellular model of organoid growth.

SAN model

We now consider a simplified, yet biologically realistic stochastic cellular model of organoid growth we call the SAN model that explains the observed linage size distributions. The model distinguishes between three types of cells based purely on the proliferation behavior they exhibit (Fig 2A). Cells are either symmetrically dividing (S-cells), asymmetrically dividing (A-cells) or non-dividing (N-cells). In our model, S-cells have the ability to self-renew indefinitely through symmetric division and can thus be considered stem cells. They form the initial cell population of an organoid, and apart from dividing symmetrically they differentiate into either A- or N-cells (or are lost, i.e. removed permanently). A-cells have committed to a differentiation trajectory and produce N-cells through asymmetric division, while N-cells do not further divide.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. The SAN model.

(Experimental data from Esk et al., [5]) (A) Division and differentiation events that S-, A- and N-cells can undergo (see Table 1 for the corresponding rates). (B) Posterior distribution for the estimated rates of division and differentiation events for days 6–11 and 11–40 (events not shown are assumed to have rate 0). (C) Total organoid size (number of cells) observed experimentally (black) and predicted by the SAN model (red), and the predicted number of S- (blue), A- (yellow) and N-cells (green). Raw data available in S4 Table. (D) Rank-size distributions observed experimentally (black) and predicted by the SAN model plus a model of NGS-based lineage tracing (red). (E) Experimentally observed number of lineages (black), predicted number of extant lineages (dotted red) and predicted number of observed lineages (red; based on the SAN model and a model of NGS-based lineage tracing, see methods “Simulation of NGS- based lineage tracing” for details). Number of obs. lineages in each replicate listed in S3 Table.

https://doi.org/10.1371/journal.pcbi.1012054.g002

We categorize cells into S, A and N solely based on proliferation behavior, not by functional categories such as for example radial glial cells (RGCs). We furthermore only consider such divisions as symmetric that produce offspring cells which are functionally identical to the parent. This precludes divisions which may appear symmetric, but where one of the offspring in fact has an altered cellular state which puts it on differentiation trajectory that leads to eventually asymmetrical division or non-proliferation. While appearing symmetric, such divisions are in the context of the SAN model more appropriately modelled as S to A transitions, since the number of eventual N-cells resulting from such a division is relatively well-determined (yet not necessarily fully deterministic).

In the SAN model, all division and differentiation events occur randomly and independently for each cell with specific time-dependent rates (Table 1). From a single-lineage perspective, the SAN model is thus stochastic in nature. Any difference between the trajectories of the lineages arising from different ancestral cells is thus assumed to be purely the result of random chance, not of cell fate decisions or spatial configuration. From a whole-organoid perspective, on the other hand, the SAN model is deterministic, because random effects average out over the roughly 10,000 lineages comprising an organoid.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Division and conversion rates in the SAN model.

Rates specify the number of expected events per cell and day. These rates are the maximum a-posteriori (MAP) estimates, the full posterior distribution is shown in Fig 2B.

https://doi.org/10.1371/journal.pcbi.1012054.t001

Division and differentiation rates

To find the rates of cell division and differentiation, we split the organoid development into four time-intervals (days 0–3, 3–6, 6–11 and 11–40) according to the main phases of the protocol of Lancaster et al. [4]. Until day 6, formation of embryoid bodies (EBs) is still ongoing, and organoid development thus does not reflect development in vivo. For these time intervals we manually chose rates of S-cell division (S → S S), removal (S → ∅) and death (S → N) for which predicted and observed numbers of cells, lineages, and lineage sizes match experimental data (Table 1). Here, distinguishing S → ∅ and S → N is necessary since dead cells within the embryoid body (EB) are still counted by sequencing-based lineage tracing, while cells which are not part of the EB are not. After day 6, EB formation is complete, and no further cells are removed from the organoid. Until day 11 S-cells are thus assumed to either divide symmetrically (S → S S) or cease proliferation (S → N), but to not produce A-cells yet. After embedding the organoids into Matrigel droplets on day 11, organoid growth enters the asymmetric division phase where S-cells are assumed to multiply (S → S S) and to differentiate into A-cells (S → A), which then produce N-cells through asymmetric division (A → A N) before they eventually cease to proliferate (A → N). We do not consider direct differentiations of S cells into N cells here since from a perspective of lineage sizes, the occurrence of such events is difficult to distinguish from slightly elevated rates of S → A and A → N. Note that A-cells decide randomly after each asymmetric division whether to continue dividing or to cease proliferation, and the N-cell output per A-cell is thus stochastic.

From day 6 onwards organoid development reflects development in vivo [5]. To obtain a range of likely rates for each event in addition to a single most-likely value, we adopted a Bayesian approach. In a Bayesian setting, one starts with an a priori assumption about likely parameter values and updates it based on experimental observations. This yields an a posteriori distribution that reflects how likely different parameter combinations are given the experimental evidence. We assumed a priori that all division and differentiation rates between 0 and 4 events/day (on average) are equally likely, and that measurement errors follow log-normal distributions. We then sampled 1,000 parameter combinations from the posterior distribution using a Markov Chain Monte Carlo (MCMC) approach (see Methods). To verify that the MCMC algorithm had converged to the true posterior distribution, we verified that the Gelman-Rubin [10] diagnostic lies below 1.2 (S1 Table). Finally, to arrive at a single set of most-likely values for the rates to be estimated, we then computed MAP (maximum a-posteriori) estimates (Table 1) from this posterior distribution.

Both the posterior distribution (Fig 2B) and the MAP estimates (Table 1 and S6 Table) show a net rate of S-cell proliferation (the difference between the rates of S → S S and S → A) close to zero. From this, we conclude that the size of the S-cell population changes only slowly from day 11 onwards. The posterior distributions of the other rates are, on the other hand, much broader. While the MAP estimates are thus arguably the single most likely set of rates, other combinations of rates are possible as well.

Model validation

For the maximum a-posteriori (MAP) rate estimates (Table 1), the organoid sizes predicted by the SAN model between day 0 and 40 agree well with the experimentally determined number of cells (Fig 2C). Similarly, the lineage size distribution predicted by the SAN model matches the observed lineage size distribution (Figs 2D and S2). In particular, the predictions show the same truncated Zipfian distributions as the experimental data and recapitulate the spread over 4–5 orders of magnitude. The SAN model predicts the number of extant lineages (lineages containing at least one S-, A or N-cell) to drop to about ≈13,700 on day 3 where it then remains. This drop in the number of extant lineages is caused by lineages that do not make it into the organoid during EB formation. This predicted number of remaining lineages (≈13,700) slightly exceeds the experimental observation (≈10,900 on day 40 on average). Since sequencing is an inherently stochastic sampling process, this excess of predicted over observed lineages caused by extant lineages which remain unobserved and is therefore to be expected. Once we take the stochastic nature of sequencing into account (see Methods for details) and compute the number of lineages we should expect to observe instead of the number of extant lineages, the numbers match closely (Fig 2E). To confirm the close fit of model and data, we replicated the experiments performed by Esk et al. [5] using the published protocol. These replicate experiments agree well with the original data of Esk et al. and show a similarly close fit between model and data (S3 Fig).

Individual effect of each rate

To clarify the individual effect that each division and differentiation rate has on organoid size and lineage size distribution, we performed a sensitivity analysis where we modified the MAP estimate of each rate (Table 1) by ± two standard deviations of the posterior distribution (Fig 2B and S6 Table). The results show that the rates between days 6 and 11 strongly affect the shape of the lineage size distribution, but not the sizes of the largest lineage (S5 Fig). The large effect of individually modifying the rate of S → S S or S → A (S6 Fig) confirms the strong correlation of these rates, previously observed through the narrow posterior distribution of the net S-cell proliferation rate (Fig 2B).

A-cell output

According to the MAP rate estimates for the SAN model (Table 1) a single A-cell has two options, either to continue to divide asymmetrically (probability r_A→AN/(r_A→AN+r_A→N) = 91%) or to cease proliferation (probability r_A→AN/(r_A→AN+r_A→N) = 9%). Since every additional asymmetric division produces an N-cell, the likely range of additional N-cells produced over the lifetime of an A-cell is thus 0 to 30 (95% quantile; 0.91³⁰≈0.05), with an average of r_A→AN/r_A→N = 10. This range of N-cells produced by a single A-cell is comparable to the range of neurons (1–35) produced by a single radial glial cell (RGC) in mice [3].

Predicted S-cell population size

The MAP rate estimates (Table 1) predict that organoids contain ≈9,500 S-cells on day 11 and ≈5,800 S-cells on day 40. To take the inherent ambiguity of the MAP estimate due to the broadness of the posterior distribution into account, we computed the posterior distribution of S-cell population sizes (Fig 3A). We find that the total S-cell population size on day 11 shows a (relatively) sharp peak at about 10,000 cells. On day 40, the estimates are more dispersed, owing to cumulative stochastic effects and to the large effect of small changes to the rates over days. Yet while we cannot pin-point the exact population size for day 40, the posterior distribution indicates that likely values range from roughly 100 to 100,000. In particular, we find that organoids very likely contain at least some S-cells on day 40.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. The S-cell population.

(A) Posterior distribution of total S-cell population size on day 11 (left) and day 40 (right). Posterior distribution obtained with MCMC, see Methods. (B) Number of fast-growing lineages (black; from Fig 1E) number of lineages with extant S-cells as predicted by the SAN model (blue). (C) Distribution of S-cells per lineages (considering only lineage with extant S-cells) on days 11, 20, 30, 40 predicted by the SAN model. (D) Neutral competition among lineages (lineage identifiers 1,2,3) within the S-cell population. The clonal composition of the S-cell distribution changes through stochastic differentiation and replenishment events, causing some lineage to eventually lose all S-cells and others to become dominant.

https://doi.org/10.1371/journal.pcbi.1012054.g003

Fast-growing lineages contain S-cells

While the total size of the S-cell population changes only slowly, its clonal composition changes rapidly. From the ≈13,700 lineages comprising the organoid from day 3 forward, ≈1,700 lineages still contain S-cells on day 11, and at day 40 that number has dropped to ≈200 (Fig 3B). This drop in the number of lineages with extant S-cells is offset by an increase in the number of S-cells each of these lineages contains (Fig 3C; average grows from ≈5 cells/lineage on day 11 to ≈36 cells/lineage on day 40).

The number of lineages with extant S-cells matches the number of lineages classified as fast-growing by our fast-slow model well (Fig 3B). This implicates S-cells as being the main driver of lineage growth. As stated above (A-cell output) a single differentiating S-cells on average produces 10 additional N-cells, and once a lineage contains no more S-cells, its growth will thus slow down and eventually cease.

Neutral competition shapes S-cell clonal composition

In linages with extant S-cells, the average number of S-cells grows continuously. At the same time, the spread between these lineages’ S-cells counts (from about 1–30 S-cells per lineage on day 11 to about 1–300 S-cells per lineage on day 40) grows as well. The clonal composition of an organoid’s S-cell population thus grows more and more non-uniform over time. Under the SAN model, this change is the result of neutral competition (Fig 3D) amongst S-cells, a term introduced to describe the population dynamics of stem cells within intestinal crypts [1].

Qualitatively, the dynamics of an organoid’s S-cell population under neutral competition mimic the population-genetic Moran model [11] in which individuals (cells in our case) carrying different neutral alleles (lineage identifiers in our case) are randomly removed (differentiate) and are replaced (through symmetric division) by offspring of another randomly selected individual. Once the last S-cell of a particular lineage has differentiated, the lineage cannot reappear within the organoid’s S-cell population. The observed disappearance (Fig 3B) of lineages from the organoids S-cell population is thus a result of more S-cells differentiating than dividing due to random chance. Similarly, the observed growth of the remaining lineages (Fig 3C) results from more symmetric divisions than differentiations, again due to random chance.

Using the SAN model, we now study the effects of neutral competition between S-cells on the clonal composition quantitatively.

Lineage-specific S-cell extinction times determine final lineage sizes

Under the population-genetic Moran model, alleles eventually either disappear from a population or become fixed. Tissue homeostasis driven by a stem cell population under neutral competition likewise leads to eventual monoclonality, i.e. to all extant cells being eventually derived from a single ancestral stem cell [1]. In growing neural tissue like cerebral organoids however, the lack of constant cell turn-over restricts eventual monoclonality to S-cells. The clonal composition of the N-cell population instead records the evolution of the S-cell’s clonal composition over time; lineages whose last S-cell was lost later and/or which contained more S-cells will contribute more N-cells than lineages which die out quickly from the S-cell population.

To study the effects of S-cell extinction on lineage sizes quantitatively, we simulated 50,000,000 lineages using the SAN model with the MAP rate estimates given in Table 1. We then stratified the simulated lineage growth trajectories according to their S-cell extinction time (T_S; the time at which a particular lineage loses the last S-cell). Lineages whose S-cell population goes extinct at day T_S = 13 (Fig 4A left) respectively day T_S = 25 (Fig 4A middle) show diminished growth and a declining number of A-cells after losing their S-cells at time T_S. In contrast, lineages whose S-cell population survives past day 40 (Fig 4A right) grow considerably faster and reach a considerably larger size. Comparing the variations in lineage sizes on day 40 between S-cell extinction time strata shows the variation due to T_S to dominate the variations within each stratum (Fig 4B). Thus, while other random factors have some influence, their influence on a lineage’s sizes on day 40 is negligible compared to the time the lineage loses its last S-cell.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Lineage-specific S-cell extinction time determines linage size.

(Shaded areas show the range between the 2.5% and 97.5% quantile across 5 billion simulations). (A) Lineage-specific growth trajectories under the SAN model stratified by the lineage’s S-cell extinction time T_S. Plot show the most likely total lineage size (red), and number of S- (blue), A- (yellow) and N- (green) cells comprising the lineage. (B) S-cell extinction time T_S vs. final lineage size on day 40. Plot shows simulation results (solid) and the analytical approximation L(T_S) (dotted). (C) Recovering S-cell extinction times from lineage sizes on day 40. Plot shows the number of lineages reaching S-cell extinction on each day estimated using L(T_S) from experimental data (black) and simulated data (red), and the true number of such lineages according to the SAN model.

https://doi.org/10.1371/journal.pcbi.1012054.g004

Mathematical analysis of the SAN model yields the approximate expression (Eq 1) for the final lineage size of a lineage comprising S₀ S-cells on day 11 (S₀≈5 for rates in Table 1; and we assume r_S→SS ≈r_S→A) and whose S-cell population goes extinct ΔT_S days after day 11. We note that final lineage size here does not refer to the size on day 40 (or any other particular point in time), but rather to the eventual size a lineage will have reached when its growth ceases. While this does not exactly match our simulation setup (we only simulate up to day 40) the approximate final linage sizes L(ΔT_S) still matches the simulation results well (Fig 4B).

Recovering S-cell extinction times from final lineage sizes

By solving the equation L(ΔT_S) = L_i, the time at which a lineage lost its last S-cell can be estimated from the final size (L_i) of that lineage. To gauge the reliability of this approach, we applied it to a simulated lineage size distribution for day 40 and found that it recovers the number of lineages that reached S-cell extinction on a particular day well (Fig 4C). When applied to the experimentally observed lineage sizes on day 40, the estimated number of lineages reaching S-cell extinction lies close to the SAN model prediction, but slightly exceeds it up to about day 30.

Emergence of a Zipfian law

If A- and N-cells are disregarded, the SAN model is equivalent to the well-studied birth-death process. Our relationship L(ΔT_S) between lineage size and S-cell extinction time together with results about the distribution of S-cell extinction times ΔT_S [12] can be used to mathematically infer the lineage size distribution expected under the SAN model. We find that lineage sizes under the SAN are indeed expected to follow a (truncated) Zipfian law, but only provided that the S-cell population is long-lived and roughly constant in size. Furthermore, in this case the predicted value α = 0.5 for the Zipfian parameter closely matches the empirically observed value α≈0.46 (see “Mathematical Analysis” in the methods section for details). The mathematical analysis of the SAN model thus independently supports the conclusion that S-cells divide and differentiate with very similar rates that we drew earlier from the MCMC-based rate estimates (Fig 2).

Experimental data

The experimental data used comprises the lineage tracing data of Esk et al. [5], the data for the additional replicates presented in S3 Fig (see Sequencing data processing), and the total organoid sizes measured using a combination of fluorescence-activated cell sorting (FACS) and image-based measurements (see Total organoid sizes). Fig 5 depicts the flow of data through the different processing steps described below.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Computational Flowchart.

Data sources and data flow through the difference processing steps described in the Methods section.

https://doi.org/10.1371/journal.pcbi.1012054.g005

Total organoid sizes

For days 0 through 21, organoid sizes (in numbers of cells) were small enough to be measured using fluorescence-activated cell sorting (FACS). For days 11 through 40, organoid volumes were estimated from microscopy images. Organoid volumes were translated into cell counts based on the average number of cells per volume computed for days 10 from FACS and volume data (FACS data for day 10 was interpolated using days 9 and 13). The number of replicates was larger than the number of organoids eventually sequenced; S4 Table contains the raw FACS- and volume-based cell counts.

Sequencing data processing

The pre-processed lineage tracing data of Esk et al. (2020) was obtained from GEO (accession GSE151383, supplementary file GSE151383_LT47.tsv.gz), and organoids “H9-day06-03” and “H9-day09-01” removed as outliers. Based on the assumption that in all samples the most common lineage size is 1 cell, we located the mode of the log-transformed read count distribution for every sample and used it to normalize relative lineage sizes (reads) to absolute cell counts. The validity of the underlying assumption is confirmed by the good agreement between the sum of absolute lineage sizes and the FACS and area-derived estimates of total organoid size. The data for the additional replicates shown in S3 Fig was error-corrected and filtered as described by Esk et al. [5]. The additional replicates were not sequenced deeply enough to reliably detect the mode in the lineage size distribution that corresponds to single-cellular lineages. To covert relative lineage sizes (measured in number of reads) to absolute lineage sizes (measured in number of cells) for these samples, we thus estimated the number of reads per cell by comparing the library size to the total organoid sizes shown in Fig 2B. The normalized version of both the data of Esk et al. [5] and the additional replicates shown in S3 Fig is available as part of our R package SANjar (http://github.com/Cibiv/SANjar).

Pareto index and fast-slow threshold estimation

For each organoid, we used the observed lineage sizes l₁,⋯,l_n to estimate the Pareto equality index α and minimal lineage size m with the maximum-likelihood estimator and computed the steady-state average from the alpha estimates of all organoids sequenced on day 11 or later. To find the fast-slow threshold l_Th for a particular organoid, we first found intersect d^Pareto such that the Pareto-induced rank-size powerlaw fits the size of the smallest observed lineage, and determined the smallest rank R for which the actual lineage size l_(r) matches or exceeds the power law L^Pareto(r). We then fit a separate log-log-linear model log₁₀ L^Large(r) = klog₁₀ r+d^Large to lineages with ranks (which we assume are surely not governed by the Pareto law), and set l_Th to the size at which the two laws intersect (meaning l_Th = L^Pareto(r) = L^Large(r)).

SAN model

In the SAN model, the clonal composition of a neural organoid at time t is described by the number of S-, A- and N-cells in each of the lineages that comprise an organoid. We now focus on a single lineage. The state of a single lineage at a particular point in time is described by an integer-valued vector (s,a,n) where s denotes the number of S-cells, a the number of S-cells and n the number of N-cells. Cells can divide (producing new cells), differentiate (changing to a different cell type) or a removed as depicted in Fig 2A. For a particular cell, these events occur stochastically with event-specific and potentially time-dependent rates , and r_A→N. Since the SAN model is stochastic, the appropriate mathematical description is in terms of a function p(s,a,n,t) which represents the probability of the lineage comprising a certain number of S-, A- and N-cells at a specific time. Initially, at time t = 0, a lineage contains a single S-cell, and thus p(1,a,n,0) = 1 (and p(s,a,n,0) = 0 for all other values of s). For other time points, p(s,a,n,t) is, at least in principle, fully defined by a so-called master equation. The master equation is found by accounting for all ways in which the system can enter or leave a specific state over an infitesimal time interval. For the SAN model the master equation for p(s,a,n,t) is

While this master equation unambiguously defines the SAN model mathematically, it is not a convenient tool for numerical simulations. The numerical algorithm we use for simulations is described under SAN model simulation below.

The full, stochastic SAN model is relevant when we consider the behavior of individual lineages. When we consider the total cell counts of a whole organoid, however, stochastic effects average out. In that case, the so-called deterministic SAN model is useful. It describes the expected total number of S-, A- and N-cells across a large number of lineages, each governed by the stochastic SAN model. The ordinary differential equations (ODEs) for the deterministic SAN model are

SAN model simulation

The total number of S-, A- and N-cells that an organoid is predicted to comprise at time t is computed by evaluating the analytical solution of the deterministic SAN model. This is done separately over each interval in which the rates are constant. The initial number s(0) of S-cells is set to 30,000 instead of 24,000 to account for a slight excess in the number of observed lineages on day 0, likely due to a combination of multiple labelling and sequencing artefacts.

To find the predicted lineage size distribution at time t, the stochastic SAN model is simulated independently for each of the 30,000 lineages in an organoid. The simulation proceeds in discrete time steps Δt, which are chosen small enough to make the probability of a single cell undergoing two events negligible (<10⁻³). Given the numbers S_i(t), A_i(t), N_i(t) of S-, A-, N-cells comprising lineage i at time t, the number of cells Δ_e undergoing event e is chosen from a Poisson distribution. Specifically, and the number of S-, A-, N-cells at time t+Δt is then set to be

Finally, the lineage size distribution l₁,⋯,l_30,000 at time t is found by summing up the number of S-, A- and N-cells, l_i(t) = S_i(t)+a_i(t)+n_i(t).

While continuous-time methods exist to simulate the SAN model exactly, these methods have the drawback that that they simulate each event separately. Their runtime thus strongly depends on the total number of cells produced, and hence on the parameter values. Since the MCMC strives to fully explore the parameter space, continuous-time algorithms perform much worse than our discrete-time algorithm, to the extent that they make running MCMC to convergence impractical. To ensure that using a discrete-time algorithm did not affect our results, we compared our simulation results against an exact algorithm and verified that the differences were negligible.

Simulation of sequencing-based lineage tracing

The effect of PCR amplification and sequencing on the observed lineage sizes was simulated using a stochastic model of PCR amplification and sequencing [15] with parameters PCR efficiency and average reads per molecule (in our case per lineage). For every sampling time t, we simulated one read count (normalized to one read per cell on average) per lineage; parameters were PCR efficiency 35% (estimated from the day 0 data) and average reads per lineage Wl_i/Σ_il_i for a linage comprising l_i cells (W is the median experimental library size for time t). The simulated read counts where then normalized to cells by division by the average number of reads per cell (W/Σ_il_i).

SAN rate estimation

To account for the major phases of the neural organoid protocol of Lancaster et al., (2017), we sub-divide days 0–40 into the four time-intervals 0–3, 3–6, 6–11 and 11–40. Briefly, until day 3, cells are expected to acclimatize and not divide. Until day 6, embryoid body (EB) formation is ongoing, and lineages may still be lost if not all cells end up as part of the EB. Between days 6 and 11, only symmetric divisions are expected to occur. On day 11, cells start to differentiate, and asymmetric divisions start to occur.

For days 0–3 and 3–6, rates which replicate the experimental data well were found by trial and error. For the remaining 6 biologically relevant rates (of S → S S and S → N between 6 and 11, and S → S S, S → A, A → A N and A → N between days 11 and 40) we used a Markov-Chain Monte Carlo (MCMC) method to compute their posterior distribution given the experimental observations. The experimental observations comprised the total organoid sizes (on days = {0, 3, 6, 9, 10, 13, 14, 16, 17, 19, 21, 22, 25, 28, 31, 32, 35, 37, 38}), and the ranked lineage sizes (on days 11 and 40, for ranks = {1, 2, 5, 10, 15, 25, 40, 60, 100, 150, 250, 400, 600, 1000, 1500, 2500, 4000, 6000, 10000, 15000, 25000}). For the total organoid size, we included all available experimental data to strongly penalize against parameters values which yield unrealistic growth behavior. For the ranked lineage sizes, we chose to include only the endpoints of the two time-intervals (days 6–11 and 11–40). This leaves the days in between available for model validation. To account for biological differences between replicates we assumed that experimental observations are log-normally distributed. The log-likelihood of the rate vector θ (comprising the 6 rates mentioned above) given the experimental data is thus

Here, c^{(day t)} and are the SAN model predictions for total organoid size and ranked lineage sizes given parameter values θ, and μ[⋯] and σ[⋯] denote the mean respectively standard deviation across biological replicates. Rates were restricted to lie between 0 and 4 and a priori assumed to be equally probable; the posterior probability of θ is thus proportional to l(θ).

To find this posterior distribution, we sampled 1,000 random rate vectors according to their likelihoods by simulating 1,000 Markov chains using pseudo-marginal Metropolis-Hastings Markov chain Monte Carlo sampling [16–18]. Each chain was initialized with random parameters drawn uniformly from the interval [0,4], and each chain was run until it had accepted 1000 moves. To verify convergence of the MCMC chains, we computed the Gelman-Rubin (GR) diagnostic [10] for each estimated parameter (plus the net S rate, i.e. the difference between the rate of S-cell production and differentiation). Our choice of uniformly distributed initial values satisfies the requirement over being over-dispersed relative to the posterior distribution, making the GR diagnostic applicable. For all rates except A → N, the 95% confidence intervals for the GR diagnostic lay below 1.2. For the rate A → N, the upper bound of the CI was 1.24 and the point estimate 1.19 (S1 Table). We concluded that the MCMC chains have explored the parameter space sufficiently deeply for the posterior distribution to be accurate. To find point estimates of the parameter, we computed the maximal mode of the (joint) posterior distribution with the mean-shift algorithm to obtain the MAP estimates (Table 1). The code used for MCMC sampling and mode-finding is available as part of our R package.

Mathematical analysis

If we consider only S-cells, the SAN model corresponds to the well-known birth-death process [12]. We consider the diffusion approximation of this process and restrict our mathematical treatment to day 11 and later where the rates of symmetric division (r_S→SS; birth) and of differentiation (r_S→A;amounts to death since we consider only S-cells) are similar enough to be considered identical (r_S→SS = r_S→A = λ/2). The number of S-cells within a lineage at time t (where t = 0 represents day 11) is then governed by the stochastic differential equation (SDE)

Using Onsager-Machlup theory [19,20] it is possible to find the most probable trajectory of a linage that contains S₀ cells at t = 0 and loses its last S-cell ΔT_S days later [13],

On average, a lineage grows by r_S→A A-cells per S-cell and per day, and over its lifetime every A-cell will eventually produce r_A→AN/r_A→N additional N-cells through asymmetric division. Eventually, a lineage that starts out with S₀ S-cells and loses its last S-cell ΔT_S days later will thus approximately grow to size

Integration of this expression and setting λ = 2r_S→SS yields Eq (1).

For an S-cell population governed by the SDE , the probability that ΔT_S≥t, is exp (−2S₀/λt), and for sufficiently large t the probability that ΔT_S ≈t is thus approximately 2S₀t⁻²/λ. By using Eq (1) to translate this distribution of ΔT_S into a distribution of lineage sizes we find that the distribution of lineage sizes with large extinction times ΔT_S should follow a Pareto distribution with α = 1/2. A more detailed mathematical analysis of the SAN model with an emphasis on criticality can be found in Ref. [13].