Supplement

www.sciencemag.org/cgi/content/full/science.aaf7012/DC1
Supplementary Materials for
Spatiotemporal coordination of stem cell commitment during epidermal
homeostasis
Panteleimon Rompolas, Kailin R. Mesa, Kyogo Kawaguchi, Sangbum Park, David
Gonzalez, Samara Brown, Jonathan Boucher, Allon M. Klein,* Valentina Greco*
‡Corresponding author. Email: [email protected] (V.G); [email protected] (A.M.K.)
Published 26 May 2016 on First Release
DOI: 10.1126/science.aaf7012
This PDF file includes:
Materials and Methods
Author Contributions
Supplementary Text
Figs. S1 to S8
Captions for Movies S1 and S2
Other Supplementary Materials for this manuscript include the following:
(available at www.sciencemag.org/cgi/content/full/science.aaf7012/DC1)
Movies S1 and S2
Author contribution: P.R., K.R.M. and V.G. designed experiments and wrote the manuscript;
P.R. and K.R.M. performed the experiments and analyzed the data. P.R. generated the K14H2BPAmcherry mouse. K.K. and A.M.K. performed data analysis, statistical modeling and
wrote the manuscript. S.P. performed two-photon time-lapse imaging and data analysis. S.B., D.
G. and J.B. assisted with technical aspects.
Materials and Methods:
Mice
K14-H2BPAmCherry mice were generated by the Yale Transgenic Facility. K14-H2BGFP, K14actinGFP and pTRE-H2BGFP mice were obtained from E. Fuchs. K5-rtTA mice were obtained
from Adam Glick. K14-CreER, Rosa26-stop-tTA and Rosa26-mTmG mice were obtained from
Jackson Laboratories. All procedures involving animal subjects were performed under the
approval of the Institutional Animal Care and Use Committee (IACUC) of the Yale School of
Medicine.
Experimental Treatment of Mice
For clonal labeling of IFE cells Cre activation was induced with a single intraperitoneal injection
of Tamoxifen (1µg/g in corn oil) at ~ Postnatal day (p50) or otherwise specified. For the label
retention experiments, mice with the pTRE-H2BGFP allele were given Doxycycline (1mg/ml) in
water at times specified. Preparation of the skin for intravital imaging was performed as
described recently (25). Briefly, mice were anesthetized with IP injection of ketamine/xylazine
(15mg/ml and 1mg/ml, respectively in PBS). The ear and tail epidermal areas were shaved using
an electrical shaver and depilatory cream (Nair). After marking the area to be image for
2
subsequent identification with a micro-tattoo, mice where returned to their housing facility. For
subsequent revisits the same mice were processed again with injectable anesthesia. The ear,
plantar and tail epidermal regions were briefly cleaned with PBS pH 7.2, mounted on a custommade stage and a glass coverslip was placed directly against the skin. Anesthesia was maintained
throughout the course of the experiment with vaporized isofluorane delivered by a nose cone.
In vivo imaging and clonal analysis
Image stacks were acquired with a LaVision TriM Scope II (LaVision Biotec, Germany)
microscope equipped with a Chameleon Vision II (Coherent, USA) 2-Photon laser. For
collection of serial optical sections a laser beam (940nm for GFP and 1040nm for mCherry,
respectively) was focused through a 20X or 40X water immersion lens (Zeiss W-PlanAPOCHROMAT, N.A. 1.0; Zeiss W-LD C-APOCHROMAT, N.A. 1.1 Zeiss) and scanned with
a field of view of 0.5 mm2 at 600Hz. z-stacks were acquired in 2µm steps for a ~30-50 µm range,
covering the entire thickness of the epidermis. Clonal analysis was performed by re-visiting the
same area of the epidermis in separate imaging experiments. A micro-tattoo was introduced in
addition to using inherent landmarks of the skin to navigate back to the original region; including
the vasculature and distinctive clustering of hair follicles. Single clones were identified during
the first imaging session (Day 0), two days after Tamoxifen induction or otherwise specified. In
subsequent sessions the same clones were traced and changes in their proliferative or
differentiation state were documented for each time point. In mice carrying both K14-CreER and
Rosa26-mTmG alleles rare recombination events were observed in the absence of induction,
indicating leakiness. Such non-specific clones were easily identified and excluded from the
analysis.
3
Photo-activation
Photo-activation in K14-H2BPAmCherry mice was carried out with the same optics as used for
acquisition. An 810 nm laser beam was used to scan the target area (10–500 mm2) and activation
of the PA-mCherry was achieved using 3% laser power for 2 min.
Image analysis
Raw image stacks were imported into Fiji (NIH) for further analysis. Images and Supplementary
Videos typically represent single optical sections selected from the z-stack, unless otherwise
specified. For visualizing individual labelled cells expressing the pTRE-H2BGFP and mTmG
Cre reporter, the brightness and contrast were adjusted accordingly for the green (mGFP) and red
(mTomato) channels and composite serial image sequences were assembled as previously
described (25).
Summary of Statistical Analysis
All details on analysis of lineage trees are provided in the Supplementary Theory and Data
Analysis (STDA) section later in this supplement. In the main text, data are expressed as
percentages or mean ± SEM. An unpaired Student's t-test was used to analyze data sets with two
groups. For all analyses, p < 0.05 was accepted as indicating a significant difference. Analysis of
lineage trees to extract lifetime correlations and fate correlations was carried in MATLAB. Pvalues for Pearson correlations on the lineage tree were calculated by the t-test using the
MATLAB corrcoef function.
4
Supplemental Figures:
Figs. S1 to S8
Fig. S1. Fluorescent reporters for in vivo lineage tracing
(A) Schematic for in vivo pulse chase system for single-cell label retention approach. 2-month
old mice were first imaged 2 days after tamoxifen induction and were then continuously treated
with Doxycycline to stop the expression of the H2BGFP reporter using K14-CreER; Rosa26-
5
mTmG; Rosa26-stop-tTA; pTRE-H2BGFP. (B) Representative optical sections of the different
layers of the interfollicular epidermis. (C-D) Examples of cell differentiation and cell division
events respectively. Scale bars: 20 µm.
6
Fig. S2. Basal cell behavioral kinetics in plantar and tail epidermal regions
(A) Single cell fate tracking by single-cell label retention. 1-month old mice were first imaged 2
days after tamoxifen induction and were then continuously treated with Doxycycline to stop the
expression of the H2BGFP reporter using K14-CreER; Rosa26-stop-tTA; pTRE-H2BGFP. The
fate and subsequent behavior of identified cells was determined by live imaging on daily revisits.
Representative time sequence of a single-cell label retention experiment showing temporal
coordination of sibling stem cell behavior. (B) Distribution of clone size at two days after
labeling in plantar epidermis, compared with the simulation based on the division/differentiation
kinetics obtained after two days. (C) Empirical and fitted lifetime distributions of dividing and
differentiating cells (n = 142 and 148 cells in ear; n = 100 and 102 cells in paw, respectively).
7
The fits reveal a significant refractory period in the lifetimes of both dividing and differentiating
cells. (D) Summary of the average lifetimes and refractory periods estimated for various
epidermises. The values were obtained by fitting the lifetime distributions for the ear and plantar
epidermises (see Fig. 1E and ST S-2.1.2) and by fitting the survival probability distributions for
the tail epidermis. (E) Survival fraction of dividing and differentiating cells in the tail epidermis
obtained from photo-activated single cells. Fitted by theoretical curves (ST S-2.2) to estimate the
average lifetimes and refractory periods. Scale bar, 25µm.
8
Fig. S3. Basal cell behaviors fit a stochastic model with lifetime refractory periods
(A) Zero-parameter fits of the stochastic cell fate model to the clone size dynamics, showing the
trajectories of individual clones (solid lines) alongside theoretical histograms (color maps)
obtained from the “non-committed progenitor” cell model (see ST S-3). These plots show the
same data and theory used in plotting Fig. 1E, but now following individual clones rather than
population statistics. Plots (top left to bottom right) show clones that disappear from the basal
layer respectively at days 4, 6, 8, or 10 post-labeling, or survived until after 10 days. (B)
Goodness-of-fit of the non-committed progenitor model with (Fig. 1F) and without the temporal
correlations between siblings (ST S-3.2), measured by the Kullback-Leibler (KL) divergence
between the model distributions and the empirical distributions. Lower values, indicating better
fits, were obtained for the model compared with a normal distribution with mean and variance
measured from the data. Calculation of the K-L divergence is detailed in ST S-3.2.3.
9
Fig. S4. A photo-activatable reporter for unbiased labeling of epidermal cells
A transgenic mouse was engineered with a K14-H2BPAmCherry allele. The H2BPAmCherry
reporter is ubiquitously expressed in all basal epidermal cells but does not fluoresce in its native
state. Upon activation by laser scanning with the femtosecond Ti:Sapphire laser the reporter
irreversibly fluoresces in the red spectrum when is subsequently excited. Groups or even single
epidermal cells can be labeled by photo-activation and traced. The K14-actinGFP allele may be
used as a ubiquitous epidermal reporter in scheme. Scale bars: 50 µm.
10
Fig. S5. Evaluating cell fate using the photo-activatable reporter
(A) Individual cells throughout the epidermis may be selectively labeled and traced with the
K14-H2BPAmCherry reporter. (B) Quantification of the percentage of dividing and
differentiating cells following single cell photo-labeling. n=103, 2 mice. (C) Representative
example of tracing five neighboring cells specifically labeled either in the first differentiated
layer (Spinous; top panels) or the basal undifferentiated layer of the IFE (bottom panels) over
one day. Scale bar: 20 µm.
11
Fig. S6. Global label retention reveals comparable behavioral kinetics in all basal cells
12
(A-C) Tracking of global epidermal turnover rate by label retention in the ear. 2-month old K5tetOFF; pTRE-H2BGFP transgenic mice were first imaged. Upon addition of Doxycycline to
stop the expression of the H2BGFP reporter the same regions in the same mice were revisited
one and two weeks after. Z stacks spanned the whole skin epithelium. Epidermal basal cells are
displayed in A and B, bulge stem cells are displayed in C. Note that GFP signal is saturated in
Week 0 to show detectable signal by Week 2. (D) Raw GFP signal intensity of single cells in the
basal layer and the bulge. (E) Decay of GFP signal intensity over time in ear epidermis. The
intensities were normalized by subtracting the background and dividing by the fitted decay curve
of the GFP intensity in the bulge. The final normalized intensity was set as 1 at day 0. All fits
assumed exponential decay with a single fitting parameter (decay rate). (F) Division rates from
the GFP decay estimated by assuming division rate = GFP decay rate / ln(2), compared with the
direct estimate obtained from the lineage tree analysis.
13
Fig. S7. Vertical organization is maintained throughout epidermal differentiation
Patterned labeling and tracing using the photo-activatable K14-H2BPAmCherry reporter reveals
that epidermal cells retain their vertical topology through the process of self-renewal and
differentiation. Scale bars: 50 µm.
14
Fig. S8. Committed cells transit through pre-existing epidermal differentiation units.
(A) Revisits of a single plane of the top granular layer of the epidermis depict the majority of
granular cell shapes are retained by their predecessors (yellow arrow). (B-C) Representative
optical sections of the different layers of the IFE epidermis over time both x,y (B) and x,z (C).
Individual cells (numbered) were tracked throughout the differentiation process from basal
through cornified layers using K14-CreER; Rosa26-mTmG; Rosa26-stop-tTA; pTRE-H2BGFP.
(C) Cell trajectories in the z-plane (arrows) over time show progressive funneling of cells into
pre-existing epidermal units. (D) Representative example of a minority of differentiating
epidermal cells forming new units (green arrow). Scale bars: 20 µm.
Movies Captions:
15
Movie S1: Serial optical sections of the adult interfollicular epidermis using the Keratin14actinGFP and Keratin14-H2BPAmCherry fluorescent reporters after total epidermal photoactivation.
Movie S2: Serial optical sections of the adult interfollicular epidermis using the ActinGFP and
H2BPAmCherry fluorescent reporters. The H2BPAmCherry reporter was photo-activated
specifically in either granular or basal epidermal cells within the same field of view.
16
Supplementary theory and data analysis
Here we present the methods used to analyze the rates and correlations of cell division and differentiation in
empirically-constructed clonal lineage trees, and their application to the data.
Section S1 describes lineage tree construction. Section S2 infers lifetime distributions and their properties
from discrete time lapse measurements, for Fig. 2A,B, and S2C,D,E. Section S3 describes zero-parameter fitting of clone size distributions by stochastic cell fate models for Figs. 2C,S2B, and S3. Section S4 calculates
correlations in cell lifetimes on the trees for sister cells. This section addresses a particular bias in calculating
lifetime correlations from discrete time-lapse data, because cells share precisely the same birth time but their division/differentiation time is known only to within the sampling interval. This bias is shown not to significantly
affect results, so simple Pearson correlations are reliable. Section S5 calculates correlations in cell fates between
ancestors and daughter cells in lineage trees to show an absence of evidence for proliferative hierarchy.
S1
Empirical lineage tree construction
Each clone was represented by a binary tree with progenitor cells representing branch nodes and differentiating cells representing leaf nodes. All nodes were assigned a discrete lifetime, being the number of time points
between birth of the cell and its division or differentiation. In the ear experiment, cells were tracked for a fixed
period of time of up to 12 days post-labeling, and cells that neither divided nor differentiated by the end of the
experiment were excluded. The analysis in section S2 corrects for the experimental bias against such cells with
longer lifetimes. For the plantar epidermis, no correction was needed as the lineage trees were constructed to a
fixed depth of two generations irrespective of the time required to reach the second generation.
S2
Obtaining cell lifetime distributions, averages and variances from discrete time lapse data
S2.1
Theory
We present here a general framework for relating discrete measurements to continuous time distributions,
which is then applied to analyze the lifetimes of dividing cells, differentiating cells, cells in the spinous layer and
cells in the granular layer.
We start by defining the empirical data. From clonal tree data or from tracing single cells through supra-basal
layers, we obtain the number of cells with a lifetime that lasts for m experimental acquisition frames, which we
denote as Im . The precise definition of ‘lifetime’ depends on the particular cell type being studied, but in all
cases the theory that follows is the same. For basal cells, the lifetime is defined as the time between cell birth and
cell division/differentiation, with the latter detected as a migration event into the supra-basal layer. For suprabasal cells, the lifetime is defined as the residency time within the layer. In the ear experiments analyzed, the
time interval between observations was two days, therefore m = 0, 1, ..., M corresponds to 0, 2, ..., 2M days of
intervals. In the plantar epidermis, the time interval was one day.
Without any analysis it appears that lifetime distributions should only be measurable to within an error of ±2
days in the birth time of a cell, and ±2 days in the division/differentiation/loss time of a cell. However, a statistical
treatment results in much more accurate estimates of lifetime distributions from the same discrete time lapse data.
The general goal is to relate between the empirical data, Im , and properties of the underlying lifetime distribution
that would have been obtained from continuous-time measurements, which we denote by the probability P (τ )dτ
of a cell having a lifetime τ to τ + dτ .
S2.1.1
General relationship
To make contact between Im and P (τ ), we first define the empirical quantity Pm that corrects for an inherent
bias in Im against long lifetimes, owing to the finite duration of the experiments. Pm is the unbiased probability
of a lifetime lasting for m frames, estimated from the data to be
Pm =
Im
.
Z(N − Nm )
(1)
17
for the ear epidermis data. Here, N is the total number of cell birth events in the entire data set, and Nm is the
number of cells that were born less than m + 1 frames before the last acquisition time frame in the experiments.
Therefore, N − Nm is the total number of cells!
that were born early enough to survive for m experiment points.
M
The normalization factor Z is introduced to set m=0 Pm = 1, assuming Pm>M = 0. For the plantar epidermis
data, we simply set Pm = Im /N since the labeled cells were all traced up until differentiation or a second
division.
The number of frames that a cell lifetime covers, m, is a function of the lifetime of the cell, which we shall
denote as τ , and the time difference between the birth time of the cell and the last experimental acquisition point
that preceeds it, denoted l. Both τ and l are stochastic variables. Let P (τ ) and P (l) be their respective probability
density functions. We can write down the joint probability density of m, τ and l as
1
P (m, τ, l) = P (m|τ, l)P (τ )P (l) = δm,[[(τ +l)/L]] P (τ ) .
L
(2)
Here, L is the time difference between experiment points (L = 2 days for ear, L = 1 day for plantar epidermis
data), δn,m is the Kronecker delta, and [[x]] denotes the largest integer that is smaller than x ([[x]] = floor(x)).
P (m|τ, l) is the probability of m given τ and l. We used the general assumption P (l) = 1/L, i.e. events are
asynchronous across the tissue and independent of observation. Marginalizing over l we obtain P (m|τ ), which
is the probability of m conditioned on τ :
" L
"
P (m, τ, l)
1 L
P (m|τ ) =
dl
=
dlδm,[[(τ +l)/L]]
P (τ )
L 0
0
⎧τ
⎪
+ 1 − m for m = [[τ + L]]
⎪
⎨L
τ
=
(3)
m+1−
for m = [[τ ]]
⎪
L
⎪
⎩
0
otherwise
Pm is related to the above probability by the chain rule,
" ∞
Pm =
dτ P (τ )P (m|τ ),
0
=
⎧" L
L−τ
⎪
⎪
dτ
P (τ )
⎨
L
0
" mL
" (m+1)L
τ − (m − 1)L
(m + 1)L − τ
⎪
⎪
⎩
dτ
P (τ ) +
dτ
P (τ )
L
L
(m−1)L
mL
for m = 0
(4)
for m ≥ 1.
This completes the relationship between P (τ ) and Im through Eq. (1).
S2.1.2
Relating continuous and discrete lifetime distributions for refractory stochastic lifetimes
Following reference (16), we consider an exponential distribution with a refractory period τR and an average
lifetime τR + 1/γ (Fig. 2B):
'
0
(t < τR )
P (τ ) =
(5)
γeγ(τR −t) (t ≥ τR ).
In this model, the refractory period τR could reflect the time between cell birth and cell fate commitment, or the
time between cell fate commitment and completion of division/differentiation, or both of these combined. For
this model probability density function, we have for the case where τR < L,
⎧
L − τR
1 − eγ(τR −L)
⎪
⎪
⎪
−
(m = 0)
⎪
⎪
γL
⎪
⎨ L
γτR
γτR
−γL 2
τR
1−e
e (1 − e
)
Pm =
(6)
+
+
(m = 1)
⎪
L
γL
γL
⎪
⎪
γ(τ
−(m−1)L)
−γL
2
⎪e R
(1 − e
)
⎪
⎪
⎩
(m ≥ 2).
γL
18
whereas, when L ≤ τR < 2L,
Pm
⎧
0
⎪
⎪
⎪
⎪
⎪
2L − τR
1 − eγ(τR −2L)
⎪
⎪
−
⎪
⎨
L
γL
τR − L 1 − 2eγ(τR −2L) + eγ(τR −3L)
⎪
+
⎪
⎪
L
γL
⎪
⎪
⎪
γ(τ
−(m−1)L)
−γL 2
R
⎪
e
(1
−
e
)
⎪
⎩
γL
=
(m = 0)
(m = 1)
(m = 2)
(7)
(m ≥ 3).
These results will be applied to data in section S2.3.1.
S2.1.3
Average and CV of lifetimes
To relate the average lifetime, E[τ ], to measurements In , we observe that Eq. (3) gives,
L
∞
%
&
mP (m|τ ) =
m=0
&
=
=
L
dl
0
∞
%
nδn,[[(τ +l)/L]]
m=0
L
dl
0
τ.
''
τ +l
L
((
(8)
(9)
(10)
Therefore,
L
∞
%
nPm
=
m=0
=
&
&
∞
dτ P (τ )
0
(11)
nP (m|τ )
m=0
∞
0
∞
%
(12)
dτ P (τ )τ ≡ E[τ ]
This means that by obtaining Pm , we can estimate the average lifetime:
E[τ ] = L
M
%
(13)
mPm .
m=0
Note that this result holds true without assuming any knowledge of the form of P (τ ). The error of this estimate
can be obtained by
)
*M
*M
2
2
m=0 (mL) Pm − (
m=0 mLPm )
(Error of mean) =
.
(14)
N −1
*∞
*M
Here we set m=0 = m=0 following the assumption that Pm>M = 0.
To consider the variance estimated from the discretized lifetime distribution Pm , we shall note that by using
the variance of mL under the condition of τ
+ ∞
,2
∞
∞
%
%
%
2
Varτ [mL] :=
(mL) P (m|τ ) −
mLP (m|τ ) =
m2 L2 P (m|τ ) − τ 2 > 0.
(15)
m=0
m=0
m=0
we can write
Var[mL] − Var[τ ]
=
∞ &
%
m=0
∞
0
2
2
2
dτ [m L − τ ]P (m|τ )P (τ ) =
&
∞
dτ Varτ [mL]P (τ ) > 0.
(16)
0
19
Therefore, the variance of mL is always larger than that of the actual lifetime τ . Recalling that the mean of mL
was equal to that of τ [Eq. (12)], it follows that the coefficient of variation (CV) of mL is always larger than that
of τ . For example, for an exponential distribution [Eq. (7) with τR = 0], we can calculate the CV explicitly:
"
"
!
1
eγL + 1
γL/2
CVexp (γ, L) = γ Var[mL] − 2 =
γL γL
−1= 2
− 1,
(17)
γ
e −1
tanh(γL/2)
which is ≥ 1.
S2.2
Survival curve of single labeled cells
The founder cells of each clone are labeled and thus not tracked from birth, so their measured lifetime distribution will in general differ from that of their daughters. The lifetime distributions are identical if and only if
basal cells have exponentially-distributed lifetimes. However, we find that the lifetimes of basal cells between
birth and division/differentiation are not exponentially distributed, so we relate here between the clone founder
cell lifetime distribution and that of their daughters.
At steady-state, the probability to randomly sample a cell which has a lifetime of τ ∼ τ + dτ should be
P (τ )τ dτ
Psamp (τ )dτ := # ∞ ′
.
dτ P (τ ′ )τ ′
0
The survival probability of a randomly chosen cell after time t is then
$ ∞
τ −t
Psurv (t) =
dτ Psamp (τ )
τ
#t∞
dτ
P
(τ
+
t)τ
0#
=
.
∞
dτ P (τ )τ
0
In the case where P (τ ) is the exponential distribution with a refractory period [Eq. (5)], we have
⎧
γt
⎪
⎨1 −
(t < τR )
γτR + 1
Psurv (t) =
1
⎪
⎩
eγ(τR −t) (t ≥ τR ).
γτR + 1
(18)
(19)
(20)
(21)
Fig. S2D,E shows Equation (21) fit to the survival fraction of the tail epidermis cells labeled by the photoactivatable reporter.
In cases where only one generation of cells is tracked, from Eq. (20) one may prove that there exists an upper
bound on the average lifetime of the entire cell population based on the fraction of single labeled cells surviving
after a time t, namely,
(Average lifetime) ≤
t
.
1 − Psurv (t)
(22)
Note that there is no general lower bound.
S2.3
S2.3.1
Data analysis
Fitting continuous lifetime distributions to the discrete data
The distribution Pm was estimated from the clonal tree data using Eq. (1) for dividing and differentiating cells
in the basal layer. We show in Fig. S2C the distributions Pm obtained for dividing and differentiating cells. These
data were fitted by Eq. (7) using least-mean-squares to estimate the two parameters of the lifetime distribution,
γ and τR . Results are shown in Table S1 indicating that the data is consistent with a significant refractory period
in the lifetimes of dividing, differentiating, and spinous layer cells.
20
Ear
Plantar
Dividing
Differentiating
Spinous layer
Granular layer
Dividing
Differentiating
Average lifetime (τR + 1/γ)
2.58 days
2.60 days
1.98 days
1.90 days
2.56 days
1.36 days
Refractory period (τR )
0.91 days
0.83 days
1.30 days
0.39 days
1.18 days
0.56 days
Table S1: Parameters of model distributuion Eq. (7) obtained by fitting.
S2.3.2
Estimating lifetime averages and CVs
Using the empirical values of Pm obtained in the previous section, from Eq. (13) we directly obtained the
average lifetimes of dividing, differentiating, spinous layer, and granular layer cells (Table S2). We also obtained
the lifetime CV from Pm for each case, which are to be compared with the CVexp of an exponential distribution,
calculated by Eq. (17) using the corresponding average lifetimes. The values of the average lifetimes were consistent with those obtained in Table S1, and the differences between CV and CVexp indicated that the distributions
are non-exponential, consistent with the fits showing a significant refractory period. (Note that CVexp > 1 for
the exponential distribution due to discrete sampling, as discussed above).
To formally test whether observed lifetimes could be explained by a simple exponential distribution corresponding to memoryless stochastic lifetimes, we calculated Pm for the exponential distributions using Eq. (7) by
setting τR as 0 and using the inferred average lifetimes. We could then directly compare between these discretized
exponential distributions and the empirical distributions using the Kolmogorov-Smirnov test (p-value). Results
are given in the following table, highlighting the fact that only the Granular layer cell lifetimes are consistent with
a memoryless process and no refractory period.
Ear
Plantar
Dividing
Differentiating
Spinous layer
Granular layer
Dividing
Differentiating
Average lifetime
2.45±0.14 days
2.51±0.15 days
2.05±0.18 days
1.79±0.25 days
2.56±0.10 days
1.36±0.08 days
CV
0.67 ±0.12
0.71±0.13
0.56±0.22
0.85±0.25
0.40±0.10
0.61±0.14
CVexp
1.05
1.05
1.08
1.10
1.02
1.09
p-value
2E-4
5E-4
0.016
0.996
3E-5
9E-3
Number of events
142
148
40
38
100
102
Table S2: Average and CV of lifetimes directly obtained from Pm .
S2.4
Estimating fraction of undivided/undifferentiated cells from short time lapse data
The expected fraction of divisions and differenations occuring for a randomly labeled basal cell after two
days (Fig. S5B) was obtained from Eq. (20). The results are shown in Table S3. We compared with this value,
which is obtained from the K14-CreER labeling experiment for the ear epidermis, with the value obtained from
the photo-activation experiment (second column); there is no evidence of bias between the different labeling
methods.
We also show in Table S3 the upper bounds of average lifetimes of dividing and differentiating cells for the
ear epidermis, obtained solely from the fraction of divided and differentiated cells at day 2 in the photo-activation
experiment.
21
Divided
Differentiated
Fraction after two days
0.32±0.03
0.34±0.03
Estimate
0.33
0.32
Upper bound on average lifetime
4.2 days
4.0 days
Table S3: Fraction of randomly labeled basal cells that divided or differentiated after two days in the ear epidermis.
S3
Fitting clone size dynamics with two stochastic models of cell fate
S3.1
S3.1.1
Model definitions
The Committed Progenitor (CP) cell model
A tested phenomenological model for the dynamics of epidermal stem cells is the committed progenitor model,
see reference (14). This model has been previously discussed in detail, but the current implementation of the
model necessarily differs from that in reference (14) owing to the new discovery of non-exponential and correlated
lifetimes in this paper, and so for completeness we redefine the model here. The model assumes that there are
two types of cells in the basal layer which we call P (progenitor) and D (differentiating), and a differentiated cell
in the suprabasal layer which we call X. The P cell is fated to divide, and the D cell is fated to differentiate. A
stochastic process can be constructed to describe changes in the number of cells in a clone, with the following
rules,
⎧
⎪
⎨PP (Prob. r)
λ
P −
→
(23)
PD (Prob. 1 − 2r)
⎪
⎩
DD (Prob. r)
D
Γ
X.
−
→
Here, λ and Γ are the rates of division and differentiation, respectively (i.e. inverses of the average lifetime).
In the simplest implementation of the model, the probability distributions for the lifetimes are exponential
(memoryless) (14), corresponding to a simple Markov process that can be solved using a Master Equation. However this study shows that the distributions of the dividing and differentiating cells are non-exponential, likely
reflecting a minimum time required for cells to either divide or differentiation after committing to their fate. We
implement this non-Markovian model in a variation on Ref. (16).
S3.1.2
A non-committed progenitor (NCP) cell model
An alternative formulation of the stochastic fate model is proposed here, with emphasis on correlated sister
cell fates. All basal layer cells are assumed to commit independently to either divide or differentiate. Sister
cells independently choose their fates, but under a shared condition, which we denote by ν. The condition ν
may represent a local environment including cell and cytokine densities, or an internal property such as the
concentration of protein factors inherited from the parent cell. We can simplify the model by assuming that the
entire basal layer consists not of two cell types (P and D), but instead of a single non-committed progenitor cell
type capable of division or direct differentiation,
P −
→
P
−
→
PP
[prob. p(ν), lifetime τp ]
X
[prob. 1 − p(ν), lifetime τd ].
(24)
Again, in implementing this stochastic process one must account for non-exponential and correlated lifetime
distributions as described in the previous sub-section. Here, the probability of differentiation, p(ν), is set to be
equivalent between the siblings since they share the same ν. The condition for homeostasis reads
⟨p(ν)⟩ = 1/2,
(25)
22
where ⟨·⟩ denotes the average over the conditions ν at steady-state. The probability of two sister cells both
dividing (or both differentiating), r, is described as average of the square of the division probability:
r = ⟨p(ν)2 ⟩.
(26)
Thus this model behavior is mathematically equivalent to the CP cell model, but since
⟨p(ν)2 ⟩ ≥ ⟨p(ν)⟩2 = 0.25,
(27)
the simpler non-committed model only allows r ≥ 0.25.
Despite the mathematical equivalence of models (23) and (24), the two models differ in their interpretation.
In the non-committed model (24), the deviation of r from 0.25 can be interpreted as the extent of fluctuation of
the probability of differentiation across the epidermis, and it does not indicate a mechanism for symmetric or
asymmetric division. In Fig. 2B, we show the empirical value of Var(p) corresponding to Var(p) = r − 0.25.
S3.2
Model solutions by numerical simulation
To predict clonal dynamics from the models described by Eqs. (23) and (24), we used a Monte Carlo approach
as described in Ref. (16) to acquire statistics from 5 · 104 simulated clonal outcomes starting with a single P
type cell for the ear epidermis. Unlike other studies of stochastic fate that have used Master Equations to solve
for clonal dynamics, here we account for the non-exponential lifetime distribution of the cells, and for sister
cell lifetime correlations, which invalidate the Master Equation approach. For simulations, we instead used the
lifetime distribution described in Eq. (5), with parameters directly inferred for the data as shown in Table S1
for division and differentiation dynamics. To generate temporal correlations in sister cells in the simulation,
we produced the lifetimes of siblings from bivariate distributions [Dukic and Maric, Phys. Rev. E, 87, 032114
(2013)] with the Pearson correlation coefficients obtained from data (Fig. 2A). No parameter fitting is carried out
since all parameter values are determined from the detailed cell lineage trees, with the numerical values shown
in Fig. 2B.
Figure 2C compares the fraction of clone sizes in the data and numerical simulation of the ear epidermis, under
the condition that an initial single labeled cell is a dividing cell. Assuming that the initial cells were randomly
labeled, the lifetime of the initial cell in the simulation was sampled from the distribution in Eq. (18). For the
plantar epidermis, we directly obtained the same parameters from the lineage trees generated from cells that
existed after two days post-labeling, and we then predicted by simulation the clone size distribution at day 2 after
following single-cell clonal labeling at day 0 (Fig. S2B).
S3.2.1
Goodness of fits
In order to quantify the goodness of the fits of the above model, we quantified the difference between the
fraction of clones, fn (t), with size n (0, 1-2, 3-4, >5 cells) at different time points t = 0, 2 ....12 days, with the
probability distributions obtained from simulation, Qn (t). We calculated the Kullback-Leibler (KL) divergence
of the fit from the data:
KL =
!
!
t=day 0,2,...,12 n
fn (t) log
fn (t)
.
Qn (t)
(28)
The KL divergence has a property that it is non-negative, and is equal to zero only when the two distributions are
identical. We show in Fig. S3B the values of KL divergence obtained from different models (different Qn (t)).
The significance between the fits of the models were quantified by obtaining theh p-value as
p-value = exp(−Ntraj |KLX − KLY |)
(29)
where Ntraj is the number of clonal number trajectories in the data, and KLX and KLY are the KL divergences
obtained from models X and Y , respectively. We found that all models fitted better than the normal distribution.
23
S4
S4.1
Sibling cell correlations
Theory: probability of synchronous division/differentiation in siblings
Siblings share not only the time frame of the birth, but also the exact timing of the birth. Therefore, even when
the two lifetimes of the siblings are independently determined, the probability to find
! the two cell differentiation
events in the same time frame, Psib , can be different from the naive expectation n Pm P"n , where Pm and P"n
are the distributions of m for cell 1 and 2, respectively. For example, in the case where the lifetime of cell
division is deterministic (i.e., P (τ!
) is a delta function), the frame at which the PP sibling divides will be perfectly
2
synchronous, Psib = 1, although m Pm
< 1.
Assuming that the lifetimes of the siblings are not correlated, we obtain the probability to find cell 1 to exit
(divide or differentiate) m frames after the birth and its sibling cell 2 to exit after m frames as
# L # ∞ # ∞
Psib (m, m) :=
dl
dτ
dτ ′ P (m, m, l, τ, τ ′ )
(30)
=
=
#
#
0
L
dl
0
L
0
#
0
∞
dτ
0
#
0
∞
0
dτ ′ P (m|l, τ )P (m|l, τ ′ )P (τ )P" (τ ′ )P (l)
dlP (m|l)P" (m|l)P (l).
(31)
(32)
Here, τ was introduced as the lifetime of cell 1, and τ ′ as that for cell 2. Note that the lifetime distributions of
cell 1 and cell 2 can be different, P (τ ) ̸= P" (τ ), as in the case of PD siblings.
The probability to find the lives of siblings to end in the same time frame under the assumption that the timings
are independently determined, can be obtained from
$
Psib =
Psib (m, m).
(33)
m
Note that if P (τ ) = P" (τ ), we have
Psib (m, m)
≥
%#
L
dlP (m|l)P (l)
0
Psib (m, m) can be explicitly evaluated by using
⎧# L−l
⎪
⎪
dτ P (τ )
⎨
#0 (m+1)L−l
P (m|l) =
⎪
⎪
⎩
dτ P (τ )
mL−l
S4.2
&2
2
= Pm
.
(34)
(m = 0)
(35)
(m ≥ 1).
Data analysis: correlation of lifetimes
Before applying the theory from section S4.1, we first calculated a simple Pearson correlation coefficient for
the lifetimes of the parent cell and daughter cells. Correlations were calculated conditional on the birth day of
the cells to correct for global fluctuation of lifetimes, by first subtracting the conditioned mean lifetime across all
clones. The obtained values with the p-values from Student’s t-test are presented in Fig. 2A.
The theory from section S4.1 was then used to validate the significance of sibling lifetime correlations, by
asking whether the probability of division or differentiation of siblings in the same acquisition frame agrees with
the null hypothesis that sibling lifetimes are independent. Unlike the Pearson correlation, this test explicitly accounts for the discrete time period of the measurement. Setting the probability distributions of the cell lifetimes
as Eq. (5) with the parameters estimated as Table S1, we calculate the probability of siblings dividing or differentiating in the same frame from Eqs. (32, 35) under the null hypothesis. Comparison to the data, shown in Table
S4, rejects the null hypothesis in all ear epidermis siblings and particularly in the differentiating siblings in the
plantar epidermis. This supports the previous analysis that there is a positive correlation between the lifetimes of
all siblings in the ear epidermis, and differentiating siblings in the plantar epidermis.
24
Ear
Plantar
PP
PD
DD
PP
PD
DD
!
Corrected ( m Psib (m, m))
0.42
0.41
0.41
0.29
0.23
0.40
!
Uncorrected ( m Pm P"m )
0.38
0.37
0.36
0.26
0.20
0.38
Experiment
0.85
0.63
0.81
0.41
0.31
0.97
p-value
6E-8
4E-4
4E-7
0.05
0.30
8E-12
Table S4: Probability of division/differentiation in synchronous frames for siblings. Corrected/uncorrected
columns show the expected probabilities of sibling division/differentiation in the same frame assuming independence, after correcting/ignoring lifetime coupling due to same birth times respectively (see section S4.1). The
p-values show a χ2 test for the experiment against the null hypothesis.
S5
S5.1
Proliferative hierarchy and mother-daughter cell fate correlations
Theory: correlation in mother-daughter cell fates in models of profliferative hierarchy
We investigated whether fate choice has memory beyond one generation. Since every mother cell is a dividing
cell, we can ask whether the fate (division/differentiation) of the sibling of a mother cell correlates with the fate of
its daughters, or more generally whether the fate of previous cell generations correlates with daughter cell fate. In
the absence of proliferative hierarchy, we expect no such correlations. A hierarchical hypothesis should show that
differentiation of a sibling cell or of any cell in an earlier generation should bias daughter cells to differentiate.
To formalize the hierarchical hypothesis, we consider the hierarchical stem/committed progenitor cell model
proposed by Mascré et al. (10), following the same notation for stochastic fate choice as in Eq. (23):
⎧
⎪
⎨SS Prob. rS
λS
S −−→
SP Prob. 1 − 2rS
⎪
⎩
PP Prob. rS
⎧
⎪
⎨PP Prob. r(1 − ∆)
λ
P −
→
(36)
PD Prob. 1 − 2r
⎪
⎩
DD Prob. r(1 + ∆)
D
Γ
−
→
X.
In this model, committed progenitor cells P are biased to differentiate with a net imbalance 2r∆ per cell cycle,
while stem cells S undergo perfectly balanced cell fate choices between self-renewal (SS) and differentiation into
committed progenitor cells (PP), as well as asymmetric divisions. In the original model (10) it was proposed that
stem cells are largely quiescent (λS ≪ λ), but the presence of quiescent stem cells is ruled out for the tissues
examined in the current study by evidence of H2B-GFP dilution. If proliferative hierarchy does exist, it does not
involve quiescent cells and requires comparable cell cycles λS ≈ λ. The following results show the signature
that such a model would have in the data.
Since stem cells in this model do not give rise to differentiating daughters (D), a dividing cell with a differentiating sibling is necessarily a P cell generated from P → PD. Similarly, if any direct ancestor cell has a
differentiating sibling then the dividing cell is necessarily a P cell. We may call all such dividing cells ‘Type I’
cells. All remaining dividing cells we call ‘Type II’ cells. These have both a dividing sibling and all ancestors
giving rise to symmetric division events. Type II cells could be either an S cell or P cell, because they could be
generated by P → PP, S → SS, S → SP, or S → PP. From this observation, one can identify the expected
fraction of dividing daughters (P) for Type I and Type II cells to be,
Dividing cell type
Type I (P)
Type II (S or P)
⟨p⟩, No hierarchy
1/2
1/2
⟨p⟩, Stem/CP model
1/2 − r∆
1/2
25
To obtain these expected values we first note that ⟨p⟩ = Prob(PP) + 0.5Prob(PD) = 1/2 − r∆ for P cells, and
thus for Type I cells. For Type II cells, we assume representative probability of sampling a P or S cell, and thus
⟨p⟩ = (1 − fS )(1/2 − r∆) + fS , where fS is the fraction of dividing cells that are S cells. The steady-state
criterion for the system is fS λS = 2(1 − fS )λr∆, i.e.,
fS =
2r∆
.
2r∆ + λS /λ
(37)
With λS /λ ≈ 1 we obtain the result above.
S5.2
Data analysis: correlation in mother-daughter cell fates
We calculated the probability of division (⟨p⟩) and symmetric (PP or DD) division (2r) conditional on a
dividing cell having a sibling that is a dividing progenitor (P) or differentiates (D). Results are shown in Table
S5. For ear epidermis all clonal data was used. For the plantar epidermis, only one generation is tracked, so we
focused on clones with exactly two cells in the initial time point, and assumed that these are siblings. All obtained
probabilities were consistent with the expected values assuming no proliferative hierarchy. Therefore, there is no
evidence of fate bias that is carried on from a mother to daughter.
The nature of hypothesis testing is such that one cannot completely rule out a proliferative hierarchy as embodied by Eq. (36). This can be seen from the results in Section S5.1, where a proliferative hierarchy requires
r∆ > 0. One sees that the fate imbalance of progenitor cells is r∆ = 1/2 − ⟨p⟩ for Type I cells, giving
r∆ = 0.06 ± 0.08 in the ear and r∆ = −0.06 ± 0.12 in the plantar epidermis. These values are consistent
with r∆ = 0, in which case the fraction of S cells is 0 [see Eq.(37)] and the hierachical model Eq. (36) becomes
equivalent to the committed progenitor model [Eq. (23)].
Ear
⟨p⟩
r
Plantar
⟨p⟩
r
Fate of sibling
P
D
P
D
P
D
P
D
Expected
0.5
0.28 ± 0.02
0.5
0.34 ± 0.05
Measured
0.44 ± 0.07
0.44 ± 0.08
0.30 ± 0.03
0.22 ± 0.04
0.48 ± 0.08
0.56 ± 0.12
0.41 ± 0.03
0.31 ± 0.06
Table S5: Probability of a dividing cell to produce a dividing cell or a differentiating cell, assuming no proliferative
hierarchy. Data shows no bias, meaning that there is no evidence of mother-daughter fate correlation.
26