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The authors have declared that no competing interests exist.

Conceived and designed the experiments: WZ ZH. Performed the experiments: ZH Y-XF C-IW WZ. Analyzed the data: ZH Y-XF C-IW WZ. Wrote the paper: ZH Y-XF AJG C-IW WZ.

In multi-cellular organisms, tissue homeostasis is maintained by an exquisite balance between stem cell proliferation and differentiation. This equilibrium can be achieved either at the single cell level (a.k.a. cell asymmetry), where stem cells follow strict asymmetric divisions, or the population level (a.k.a. population asymmetry), where gains and losses in individual stem cell lineages are randomly distributed, but the net effect is homeostasis. In the mature mouse intestinal crypt, previous evidence has revealed a pattern of population asymmetry through predominantly symmetric divisions of stem cells. In this work, using population genetic theory together with previously published crypt single-cell data obtained at different mouse life stages, we reveal a strikingly dynamic pattern of stem cell homeostatic control. We find that single-cell asymmetric divisions are gradually replaced by stochastic population-level asymmetry as the mouse matures to adulthood. This lifelong process has important developmental and evolutionary implications in understanding how adult tissues maintain their homeostasis integrating the trade-off between intrinsic and extrinsic regulations.

In multi-cellular organisms, there is a static equilibrium maintaining cells of various forms. This homeostasis is achieved by an exquisite balance between stem cell proliferation and differentiation. Understanding how different species and organ types maintain this dynamic equilibrium has been an interesting question for both evolutionary and developmental biologists. Using population genetic theory together with previously published single-cell sequencing data collected from mouse intestinal crypts at two points in development, we have revealed a dynamic picture of stem cell renewal in intestinal crypts. We found that intestinal equilibrium is maintained at the single-cell level through predominantly asymmetric stem cell divisions at early life stages, but progressively switches to a population level homeostasis with only symmetric divisions as the mouse matures to adulthood. This dynamic process, likely to be conserved across species, has important developmental and evolutionary implications in understanding how adult tissues maintain their homeostasis integrating lifelong trade-offs between intrinsic and extrinsic factors.

Development and tissue homeostasis of multi-cellular organisms is an extraordinary cellular orchestra starting from a single zygote

Stem cells are a group of cells with a dual role. On one hand, they need to maintain their own population through self-renewal. On the other hand, stem cells also give rise to differentiated cells which carry out most body functions

Asymmetric division is particularly attractive and allows stem cells to accomplish both maintenance and differentiation simultaneously in a single division. However, symmetric divisions are also indispensable in situations such as morphogenesis and tissue injury where stem cells need to proliferate rapidly

Stem cells often form distributed clusters and live in local nurtured structures known as the stem cell niches

Mammalian intestine has become one of the best model systems for studying stem cell dynamics

Genomic sequencing, in particular single cell sequencing, provides a powerful alternative approach for studying cell lineage relationships. Compared to traditional molecular techniques such as the lineage tracing

Population-genetic theory

We consider a discrete-generation model of tissue homeostasis. In each cell generation, a proportion α of the cells divides symmetrically and gives rise to two descendant stem cells (type I, _{t} = (2α+β)^{t}×N_{0}, where N_{0} is the population size at time 0.

(A) Anatomical structure of the intestinal crypt. The dark green cells represent stem cells and light green cells are transit-amplifying cells. There are three types of stem cells divisions (I, II and III, see main text). (B) A cartoon illustration of a coalescent process in the two deme model. One cell from stem cell deme and two cells from the transit-amplifying cell deme were sampled. Their ancestral relationship is depicted as the gene tree connecting their ancestors. (C) The state transition for the Markov Chain in one step. The current state of the chain is (m,n) and the state in the previous generation is (m′,n′). In the example here, (m,n) = (1,3) and (m′,n′) = (2,1). (D) The expected time to the most recent common ancestor for two lineages (denoted as TMRCA2) was calculated using three different approaches. The solid line is calculated using a first-step analysis of the Markov Chain. Blue squares are the results from the forward simulation and triangles are from the direct simulation following the Markov Chain.

Now, suppose we pick two stem cells at random at time t, the probability that they will have a common ancestor in the previous generation can be computed in two steps. The first stem cell picked must be derived from the type I stem cell division in the previous generation and the probability of picking it is 2α/(2α+β). Secondly, the other sampled stem cell must be the pair of the first picked stem cell in the type I division and the probability of picking it is 1/(N_{t}−1). Thus the probability of finding a common ancestor (i.e. a coalescence) in a single generation backwards in time for two stem cells is:_{t} = N_{0} = N. Then, the above probability can be rewritten as 2α/(N−1). Once we have this single-step probability, other quantities such as the time to the most recent common ancestor for two cells and the coalescent relationship in a sample can be derived following the n-coalescent approach (

Each intestinal subunit is composed of two parts: a protrusion compartment called villus, which contains terminally differentiated cells, and an invagination compartment named crypt, which hosts stem cells and highly proliferative transit-amplifying cells. There is a continuous process that replaces functional cells in the villi with cells grown out of the crypts. We used a two-deme population-genetic model to capture continuous renewal of stem cells and transit-amplifying cells (

In practice, when we take a random sample of cells from a crypt, we do not know whether they are stem or differentiated cells. Thus, the number of sampled cells from two demes (denoted as (m, n)) will follow a hypergeometric distribution (N_{1}, N_{2}, m+n), where N_{1} and N_{2} are population sizes for the two demes. Given (m,n) cells from deme 1 and deme 2, the coalescent process of going backwards in time and finding common ancestors for these lineages can be modeled using a Markov Chain (

Given observed genotype information (e.g. microsatellite markers, denoted as D), obtained by assaying single cells, the likelihood of the data can be calculated as

In practice, given the large dimensionality of genealogical spaces, it is not feasible to exhaustively explore all possible ancestry relationships in a sample. Instead, we use a Monte-Carlo approach to compute the likelihood in _{i} is sampled from Pr(G |θ). It can be shown that, as k increases, likelihoods from

Sampled gene genealogies can be drawn either from the Markov Chain or forward simulations depending on whether a cell population has reached equilibrium (i.e. stationary distribution) at the time of data acquisition. Markov Chain calculations assume that a given population has reached equilibrium under a given configuration of symmetric/asymmetric division rates, which may not always be true. On the other hand, forward simulation can be applied to either non-stationary or stationary scenarios, but is computationally much more expensive.

Through computational simulations, we found that stationarity has been reached for samples taken on day 340 across most of the parameter space, but not on day 52 (

After averaging over many possible genealogical histories, we can calculate the likelihood of the observed data (i.e. microsatellite markers). Given the likelihood function, maximum likelihood approaches can be employed to infer the most likely parameter values. In particular, we are interested in estimating the proportion of symmetric/asymmetric divisions (α, β) in the life history of mice.

Single-cell genotype data were taken from a previous study

Time | ID | n | β |
lnL | β |
lnL |

Day 52 | Crypt1 | 6 | 0.76 | −305.12 | 0.76 (0.237, 1.0) |
−685.41 |

Crypt2 | 5 | 0.60 | −380.21 | |||

Day 340 | Crypt1 | 5 | 0.00 | −327.46 | 0.00 (0, 0.441) |
−734.29 |

Crypt2 | 4 | 0.00 | −406.82 |

: estimated for each crypt.

: estimated for each mice.

: confidence interval calculated from the non-parametric bootstrap samples.

Using a two-step mutation model for micro-satellite markers, we first calculated the genetic distances between all sampled cells (

(A) Unweighted Pair Group Method with Arithmetic mean(UPGMA) tree

Using the likelihood approach we outlined above, we calculated the likelihood of the data as a function of asymmetric division rate for each crypt (

Even though the point estimates of the asymmetric division rate is rather different, the confidence in the point estimates is not very strong. This is especially true for the day 52 (_{0}: β_{52} = = β_{340} against the alternative hypothesis H_{a}: β_{52}≠β_{340}. Since the null hypothesis is a special case of the alternative hypothesis (i.e. the models are nested), a Likelihood Ratio Test (LRT) can be employed to ask whether the null hypothesis can be rejected with confidence. After we calculated the associated test statistic, the LRT gives the p-value of 0.024, which is significant at the nominal cut-off of 5% (_{52}–β_{340} with zero over the bootstrap samples (

Model | lnL under H_{0} |
lnL under H_{a} |
−2ΔlnL | pvalue |

Two deme model (Discrete) | −1422.26 | −1419.69 | 5.13 | 0.024 |

Age structure model (Discrete) | −1442.97 | −1430.57 | 24.81 | 6.3×10^{−7} |

Spatial model (Discrete) | −1434.66 | −1425.16 | 19.02 | 1.3×10^{−5} |

Continuous time models | −1417.76 | −1415.39 | 4.75 | 0.029 |

: significant at 5% level,

: significant at 1% level.

In addition to the two-deme model outlined above, we explored a series of more complicated models that reflect various additional aspects of crypt biology. In general, due to complexity of these models, analytical results are much harder to derive, but can be supplemented with computer simulations (^{−7}). This is also true when we explore spatial structures of the intestinal crypt (P = 1.3×10^{−5},

Using population genetic theory, in particular the coalescent theory, we have drawn an extraordinary dynamic picture of intestinal crypt homeostasis. Compared to earlier lineage-tracing methods which typically do not allow for individual lineage relationships, this branch of theory provides a more detailed picture of stem-cell crypt dynamics. With single-cell data collected from different life stages, we found strong statistical support for a transition from cell asymmetry to population asymmetry during mouse life history.

Intuitively, the reason we can observe this discrepancy is that genealogical trees of crypt cell populations will steadily increase in size as the population evolves to establish equilibrium. This is analogous to the founder population effect in population genetics (

Previous observations revealed that mouse crypt morphogenesis started with a surge of symmetric divisions establishing the pool of stem cells, followed by a transition to predominantly asymmetric divisions that maintain an equilibrium between stem cell self-renewal and differentiation

(A) The proportion of asymmetric and symmetric division as a function of mouse life stage. The pattern from morphogenesis are from experimental data together with the optimal control theory prediction

The population asymmetry found here for day 340 matches previous observations that adult intestinal stem cells are maintained by replacing randomly-lost cells through predominately symmetric divisions of their neighbors (

Our observations raise a number of questions about the dynamics of the transition between cell division modes. Stem cell behaviors are often controlled by both internal signals (e.g. cellular polarity

Furthermore, transition timing is also unclear. If paneth cells are responsible for maintaining much of the intestinal niche

Interestingly, current biological evidences seem to have a tilt towards a fast transition. For example, lineage tracing study looking at the speed of drift towards monoclonality, has found similar rates for mice of age 1.5, 6.5 as well as 8 months

Why do the crypt cells need to switch to population level asymmetry, which is an apparently more fragile scheme for long-term tissue maintenance

On the other hand, there might also exist a “passive” explanation for this transition. The observed progression can simply be a by-product of natural selection. When a single gene has multiple functions, some of the functions will be beneficial to the organism, while others might be detrimental. Most importantly, when the advantageous gain outweights the deleterious costs, the target gene can still be selected (i.e. antagonistic pleiotropy)

The life-history of stem cell division is not yet fully discernable from our results (e.g.

Until now, asymmetric divisions were thought to be rarer in vertebrates than invertebrates

Since population sizes are relatively constant in the intestinal crypt and because type III divisions do not change the number of stem cell descendants, the proportions of two types of symmetric divisions (type I and II) have to be balanced and are set to be equal so that the total number of cells is maintained (

Since there is a constant extrusion of cells from the crypt into the villi, we capture the dynamics of the transit-amplifying cell pool by allowing only a certain proportion of the cells to participate in reproduction for the next generation. The remaining cells are extruded out of the deme 2. We set the population size of stem cells (population 1) to N1 and that of transit-amplifying cells (population 2) to N2. In each generation there are N1 cells migrating to the transit-amplifying cell pool. In the transit-amplifying cell population, fraction γ of N2 cells divide once and give rise to two descendant cells. The remaining (1−γ)×N2 cells are extruded outside of the population 2. The value of γ is set to (N_{2}-N_{1})/2N_{2} such that the population size in population 2 is constant. Based on previous observations in mouse colon crypts

Given the number of cells we sampled in two demes, the process of going backwards in time and finding common ancestors can be modeled as a Markov Chain. The state transitions are given by combinations of individual coalescence or migration events. For two randomly sampled lineages, the expected time to their most recent common ancestor (MRCA) can be computed by directly simulating from a Markov Chain following the appropriate state transitions. The expected value for the time to MRCA for two lineages can also be derived analytically using a first-step analysis of a given Markov Chain (

The Markov Chain calculation assumes that data are collected from a stationary process. Based on our simulations, we find that, for most of the parameter values, stationary distributions have been reached by day 340. However, this does not appear to hold for day 52 (

To estimate the number of generations leading to day 52, we tried a series of approaches. Since we know that crypt morphogenesis starts around post-natal day 7 for mice

Data were taken from a previously-published study

We also tried a series of alternative models to explore other aspects of stem cell dynamics.

In the age structure model (

In the continuous-time models, the time to the next event (waiting time) is exponentially distributed with the intensity parameter specified by the cell division rate (λ). The time to the next event for n cells is exponentially distributed with rate nλ. Given the time to the next event, the exact cell that experiences this event is randomly picked among the n cells following statistical properties of the Poisson Process

In general, analytical results from these models are much harder to derive, but computational simulations can be conducted to sample gene genealogies from the random process. Likelihood calculations follow the same procedures as previous models after sampling gene genealogies from Pr(G| θ).

Given a gene tree, the likelihood of the observed data can be computed from the tip of the tree towards the root using the pruning algorithm

In order to further explore the possibility of mutation rate variation, we used various forms of the beta distribution to capture uncertainty in the mutation rate. Since the mutation rate measured from previous studies is 0.01 per site per generation, we adopted beta distributions with different shape parameters, but transformed to take values between 0.0075 and 0.0125. The likelihood of the data can be calculated by partitioning the mutation distribution into discrete bins and taking the weighted sum of individual likelihoods calculated at discrete values of mutation rates

Unweighted Pair Group Method with Arithmetic mean (UPGMA) tree

The likelihood of the data as a function of the underlying parameters can be computed in a Monte Carlo fashion as in

The distribution of estimated β_{52}–β_{340} over the bootstrap samples. For each re-sampled bootstrap datasets, we can get an estimate of β_{52} and β_{340} respectively. When we take a difference between the two point estimates and plot its distribution, we get the histogram shown in this figure. The number of replicates with β_{52}–β_{340} less or equal to zero is 4 (out of 100).

(PDF)

Alternative models exploring additional aspects of the cellular dynamics within intestinal crypts. (A). In the age-structure model, there are multiple demes with different ages in the non-stem cells. In each generation, non-stem cells in deme i (age i) migrate into deme i+1. Non-stem cells reaching a maximum age (denoted as K) will be extruded out of the crypt in the next generation. (B). In the spatial model, multiple spatial demes exist in the non-stem cell pool. In each spatial deme, non-stem cells have a certain probability of staying in the original deme and with remaining probability of moving to the next deme or being extruded out of the crypt. The exact setup of these two models is presented in great detail in the

(PDF)

The likelihood of the data under different distributions for mutation rates. (A) Log likelihood profile for the first crypt at day 52 under different beta distributions. (B) The same plot, but for the second crypt at day 52. (C) Log likelihood profile for the first crypt at day 340. (D) The same plot, but for the second crypt at day 340.

(PDF)

Mean pairwise divergence time between two cells at different cell generations for different asymmetric/symmetric division rates. The X axis is the generation time and the y axis is the mean pairwise difference. (A) beta = 0, (B) beta = 0.2, (C) beta = 0.4, (D)beta = 0.6, (E) beta = 0.8.

(PDF)

1) The derivation for state transitions in the Markov chain. 2) Alternative models and their setup. 3) Crypt history and genealogical sampling at day 52. 4) Mutation rates and the likelihood calculation.

(DOC)

We would like to thank colleagues from Chung-I Wu's lab at Beijing Institute of Genomics for extensive discussions and critical comments. We also would like to thank the editor and two anonymous reviewers for very constructive comments and suggestions. We want to thank Prof Zhi-Ming Ma and his group for very helpful discussions through the joint program at National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences.