††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Differential elasticity in lineage segregation of embryonic stem cells

Christine M. Ritter  Niels Bohr Institute, University of Copenhagen, 2100
Copenhagen, Denmark  Natascha Leijnse  Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark  Younes Farhangi Barooji  Niels Bohr
Institute, University of Copenhagen, 2100 Copenhagen, Denmark  Joshua M.
Brickman  The Novo Nordisk Foundation Center for Stem Cell Medicine, reNEW,
University of Copenhagen, Blegdamsvej 3, 2200 Copenhagen, Denmark  Amin
Doostmohammadi  Niels Bohr Institute, University of Copenhagen, 2100
Copenhagen, Denmark  Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS>Lene B. Oddershede  Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark  Corresponding authors:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The question of what guides lineage segregation is central to development,
where cellular differentiation leads to segregated cell populations destined
for specialized functions. Here, using optical tweezers measurements of mouse
embryonic stem cells (mESCs), we reveal a mechanical mechanism based on
differential elasticity in the second lineage segregation of the embryonic
inner cell mass into epiblast (EPI) cells - that will develop into the fetus -
and primitive endoderm (PrE) - which will form extraembryonic structures such
as the yolk sac. Remarkably, we find that these mechanical differences already
occur during priming and not just after a cell has committed to
differentiation. Specifically, we show that the mESCs are highly elastic
compared to any other reported cell type and that the PrE cells are
significantly more elastic than EPI-primed cells. Using a model of two cell
types differing only in elasticity we show that differential elasticity alone
can lead to segregation between cell types, suggesting that the mechanical
attributes of the cells contribute to the segregation process. Our findings
present differential elasticity as a previously unknown mechanical contributor
to the lineage segregation during the embryo morphogenesis.

††preprint: APS/123-QED Figure 1: Autonomous segregation of the inner cell
mass occurs both inside the pre-implementation mouse embryo at day E3.5 and in
a culture of mESCs. a) Schematic illustration of the segregation inside a
mouse embryo between day E3.5 (left) and E4.5 (right) at which EPI-primed
cells (yellow) are centrally located and PrE-primed cells(red) are located at
the rim of the forming embryo. b) and c) Confocal $z$-projection of mESCs,
where b) all cell nuclei are labeled with H2B-Venus (yellow) while c) the PrE-
primed cells express an additional mCherry-Hhex fluorescent reporter (red).
Examples of EPI-primed cells which do not express the mCherry-Hhex fluorescent
reporter are marked by yellow circles in b) and c). d) Positional power
spectrum of a granule located in the cytoplasm of an EPI-primed cell. The red
line signifies the fitting range, 200-3000 Hz. Bottom inset: Schematic
illustration of optical trapping (red: trapping laser) of a lipid granule
(blue) inside the cytoplasm of an mESC (yellow). e) mESCs visualized by bright
field microscopy, lipid granules are visible throughout the cells (see red
arrows for examples). Scale bars are 20 $\mu$m.

Embryonic development is a paradigm of biophysical self-organization [1, 2].
The fertilized egg makes the early mammalian embryo or blastocyst as a result
of several rounds of cell divisions coupled to two successive lineage choices.
First, embryonic blastomeres are segregated into an inner cell mass that is
enclosed by an epithelium of trophectoderm cells. The trophectoderm will go on
to form the placenta, while the inner cell mass will give rise to the
remaining support structures and the embryo proper. The second lineage
segregation occurs when the inner cell mass differentiates into two distinct
populations: epiblast (EPI) that will develop into the fetus, and primitive
endoderm (PrE) that will form extraembryonic structures such as the yolk sac
[3, 4, 5] (Fig. 1a). Recent studies have uncovered the mechanism of the first
lineage segregation into trophectoderm and the inner cell mass based on the
coordination between differential contractility and difference in the basal-
apical polarity (pointing towards the outside of the embryo) [6], indicating
that mechanical features of early embryos play a defining role in the first
lineage decision-making [7, 8, 9]. However, the role of potential mechanical
mechanisms for the second lineage divergence into EPI and PrE remains elusive
[10]. The segregation of the inner cell mass into EPI and PrE is thought to be
driven by the interaction of FGF signaling with the gene regulatory networks
responsible for PrE and EPI identity. The propagation of this signal via a
feedback circuit leads to the induction of progenitor cells for both lineages
in a salt and pepper pattern that resolves into two separate domains based on
differential adhesion [4, 5]. While the heterogeneity of the inner cell mass
is now accepted, the differential adhesion mechanism remains controversial. In
particular, recent experiments have shown no evidence of any difference
between the main determinants of adhesion between PrE and EPI cells [10].
Similar conclusions have been drawn for epithelial cell layers, where
experiments and theory refuted cell segregation based on the differential
adhesion between epithelial cells [11]. Additionally, differential line
tension and differential affinity mechanisms have been shown to be
insufficient to govern cell sorting in the inner cell mass [12]. Here, we use
in vitro experiments exploiting PrE priming in mouse ESCs alongside with cell-
based modeling, to show that differences in mechanical properties of the cells
within the inner cell mass and in particular their distinct elastic behaviors
- rather than differences in adhesion and interaction properties - could lead
to the autonomous sorting of cells into PrE and EPI sub-populations.

Endodermal primed (PrE) cells are significantly more elastic than epiblast
(EPI) cells. Naïve embryonic stem cells are in vitro cell lines derived from
the ex vivo expansion of the inner cell mass. They are heterogeneous and can
be shown to exhibit reversible lineage priming from an epiblast-like cell to a
population biased for PrE differentiation [13]. We used mouse embryonic stem
cells (mESC) as a model system [3, 14] carrying a fluorescent mCherry-Hhex
reporter alongside a constitutively expressed nuclear label, H2B-Venus [15].
Cells exhibiting mCherry fluorescence are primed for PrE and all cells in the
culture express nuclear Venus. The degree of mCherry fluorescence within a
cell can be used to determine its priming (Fig. 1b,c) [16, 17, 14]. Using
these cells, we compared elastic properties of mCherry high and low mESCs.
Experiments are performed at E3.5, which corresponds to the “early wave” of
lineage expression and precedes the full lineage exclusive expression and
positional ordering that occur at later stages [5, 18] (Fig. 1a).

We used optical tweezers to trace the thermal fluctuations of nano-scopic
tracer particles [19, 20], in the form of endogenously occurring lipid
granules (Fig. 1d,e), to determine the local elastic properties within
individual cells [21, 22]. In the high-frequency regime, where the tracer does
not feel the restoring force from the optical trap, the power spectrum of a
diffusive tracer particle in a medium follows a power law [23] $P(f)\propto
f^{-(1+\alpha)}$, where the scaling exponent $\alpha$ carries information
about the elastic properties of the medium [24, 25, 26, 27, 28, 29]. The
closer the scaling exponent $\alpha$ is to 1, the more viscous the tracer’s
environment, the closer $\alpha$ is to zero, the more elastic the environment.
In our study, the fitting interval for obtaining $\alpha$ (200-3000 Hz) was
chosen such that it is well above 100 Hz, the frequency below which non-
equilibrium processes typically occur in biological samples, and as such
within this fitting interval we are probing the material properties inside of
individual cells [30, 31]. The scaling exponents measured for all mESCs were
significantly lower than values reported for any other cell type, showing
that, overall mESCs are highly elastic (see Fig. S1 for a comparison of the
scaling exponent to epithelial, endothelial, fibroblast, and yeast cells).
Interestingly, the average value of the scaling exponents in all PrE-primed
cells $\alpha$=$0.42\pm 0.17$ (mean $\pm$ s.d.) was significantly smaller than
the value from EPI-primed cells $\alpha$=0.49$\pm$0.13 (mean $\pm$ s.d.) (Fig.
2a and Fig. S2), indicating that the PrE-primed cells are more elastic than
the EPI-primed population at this early stage of embryonic development. This
is important because our measurements show that these observed significant
differences in the elasticity are already occurring during priming and not
just after a cell has committed to differentiation.

To further validate the results obtained from the scaling exponent analyses we
next calculated the complex shear modulus $G=G^{\prime}+iG^{\prime\prime}$ of
the lineage primed embryonic stem cells (Fig. 2c and Fig. S3) as the defining
characteristics of viscoelastic materials (see Supplementary Material for
details of the measurements) [32, 33]. The storage modulus $G^{\prime}$,
characterizes the energy stored in the material during a cycle of deformation,
and therefore is a measure of the elastic component of the material behavior,
while the loss modulus $G^{\prime\prime}$, characterizes the average
dissipation of the energy and is a measure of the fluidity of the material
[34]. Therefore, in order to assess the relative fluid-like and solid-like
behavior of the cells we measured the loss tangent
$G^{\prime\prime}/G^{\prime}$, the ratio of the loss to storage modulus for
EPI- and PrE-primed cells (Fig. 2d). Consistent with the results of scaling
exponent measurements from power spectral analyses, a significantly higher
elasticity was observed for PrE-primed cells compared to EPI-primed ones.
Together, our analyses of the complex shear modulus confirm the results of
power spectral analyses, demonstrating that (i) the mESCs are highly elastic
compared to other cell types and (ii) there is a significant difference
between the elasticity of PrE- and EPI-primed cells, the behavior of which
resemble the inner cell mass at the early stages of the embryo development
[4].

Figure 2: PrE-primed mESCs are more elastic than the EPI-primed mESCs. a)
Scatter plots of scaling exponents, $\alpha$, characterizing the sub-diffusive
motion of granules inside mESCs for PrE- and EPI-primed cells inside the
nuclei (left) and cytoplasm (right) show significantly lower $\alpha$ values
and thus higher elasticity for PrE-primed mESCs. b) Inhibiting actin
polymerization decreases mESC elasticity. Scatter plots of scaling exponents,
$\alpha$, for control cells vs. cells treated with Latrunculin B (LatB) for
EPI-primed cells in nucleus (1st plot) and cytoplasm (2nd plot) and in PrE-
primed cells in nucleus (3rd plot) and cytoplasm (4th plot). c) Complex shear
moduli, $G^{\prime}$ and $G^{\prime\prime}$, calculated for EPI- and PrE-
primed mESCs averaged over 10 different granules per experiment. d) Loss
tangent averaged over 10 different granules per experiment. The scaling
behavior of $G^{\prime}(f)$ and $G^{\prime\prime}(f)$ comply well with those
obtained from power spectral analysis in comparable frequency intervals. Both
the absolute value of $G^{\prime}(f)$ and the calculated loss tangent show
that PrE-primed cells are significantly more elastic than EPI-primed cells.
Black lines in scatter plots in a) and b) denote the medians. Mean, standard
deviations, and $p$-values for data in a) and b) are summarized in
Supplementary Tables S1 and S2.

Both nucleus and cytoplasm of primed PrEs are significantly more elastic than
in primed EPIs. Next, in order to further assess the elasticity difference
between the two sub-populations within lineage primed mESCs, we compared the
internal structure of the two sub-populations. Fluorescent labeling of the
nucleus (using a H2B-Venus expression construct) allowed for its independent
visualization. In addition, in our optical tweezers experiments, the presence
of lipid granules both in the nucleus and in the cytoplasm, allowed for
spatially resolved measurement of the elasticity locally within the cells (see
Supplementary Material for details). For both cell types, the scaling
exponents characterizing the movement of tracers in the nuclei were found to
be significantly lower than in the cytoplasm (Fig. 2a), demonstrating that for
all probed cell types, the nucleus was more elastic than the cytoplasm (see
values of $\alpha$ in Supporting Table S1 and results of $t$-tests in
Supporting Table S2). This observation is in line with a previous study of
articular chondrocytes using micro-pipette aspiration, which showed that the
nuclei were significantly stiffer than the cytoplasm [35]. Comparing the
spatially resolved scaling exponents between the nuclei of PrE- and EPI-primed
cells, we found that both the nuclei and cytoplasm of the primitive PrE-primed
cells were significantly more elastic than the EPI-primed cells. Together,
these results show that the observed differential elasticity between the mESCs
primed for EPI and the PrE is consistent for comparison of cytoplasm, nuclei,
and of entire cells.

Higher levels of polymerized actin in PrE-primed cells are linked to their
higher elasticity compared to EPI-primed cells. The difference in elasticity
of EPI- and PrE-primed cells may be attributed to the difference in structure
and composition of their subcellular elements. Although actin is predominantly
present at the periphery of the cells [36], in ESCs it is also present in the
nucleus where it has been shown to associate with lamin A and chromatin
remodeling complexes [37]. We hypothesized that a higher level of polymerized
actin filaments in the PrE-cells may produce the difference in the elasticity
between EPI- and PrE-primed cells. To test this hypothesis we treated cells
with the actin disruptive drug, Latrunculin B (LatB) (see Supplementary
Material for the details of drug treatments). As expected, for all
investigated categories (nucleus or cytoplasm, EPI- or PrE- primed), the cells
became less elastic (showed a higher value of the scaling exponent $\alpha$)
upon LatB treatment, consistent with removal of actin filaments (Table S1,
Table S2). Importantly, however, while actin disruption led to a small
decrease in the elasticity of the EPI-primed cells, PrE-primed cells became
significantly less elastic (Fig. 2b, Fig. S4, and Table S1), indicating that
higher levels of actin filaments in these PrE-primed sub-populations governs
their higher elasticity. Together, these results suggest that the difference
in internal molecular composition and structure of the primed cells lead to
the difference in their elasticity, showing that changes in internal
composition and structure already occur during priming and not just after a
cell has committed to differentiation.

Figure 3: Differential elasticity is an active mechanism for cell segregation.
a,b) Time series snapshots of the segregation of a 20-cell aggregate of soft
(yellow) and stiff (red) cells simulated a) without any confining boundaries,
and b) in a circular confinement. c) Cell segregation as a function of the
normalized elasticity difference $(\gamma_{A}-\gamma_{B})/\gamma_{B}$, where
$\gamma_{A}$ denotes the elasticity of the stiff cells and $\gamma_{B}$ is the
elasticity of the soft cells. e) Cell segregation as a function of the
strength of active stresses generated by the cells. The segregation index
$SI=\langle N_{A}/(N_{A}+N_{B})\rangle$ is defined such that $SI=1/2$ when the
outer layer is a uniform mixture of stiff and soft cells (indicated by the
dashed red line), $SI=1$ when the outer layer is occupied solely by stiff
cells, and $SI=-1$ if only soft cells occupy the outer layer. Error bars
indicate the variance computed from averaging over 5 different simulations
with random initial positioning of the cells.

Minimal physical model shows elasticity difference between PrE- and EPI-primed
cells is sufficient for the observed cell segregation. When EPI and PrE primed
mESCs are injected into host embryos they colonize their respective lineage
and when mixed in aggregation culture, they segregate, recapitulating the
differential localization of PrE and EPI in vivo [38]. Could the difference in
elasticity alone be sufficient for cell segregation? In order to answer this
question, we developed a cell-based model of the inner cell mass composed of
two cell types that only differ in their elastic properties. We build on a
phase-field model [39, 11] that captures the mechanical properties of the
cells, including elasticity, deformability and contractility, at the single
cell level and allows for accurate representation of boundaries between the
cells [40, 41, 42] (see Supplementary Material for detailed description of the
model). Importantly, this model allows us to vary the elasticity, while
explicitly keeping the same interaction forces, i.e., adhesion and repulsion
forces, between the cells. In brief, each cell is represented by a phase-field
parameter $\Phi$ that distinguishes the interior $\Phi_{i}=1$, and the
exterior $\Phi_{i}=0$, of an individual cell. The dynamics of each phase-field
evolves according to the Cahn-Hilliard equation:

$\displaystyle\partial_{t}\Phi_{i}+\bm{v_{i}}\cdot\nabla\Phi_{i}=-\frac{\delta\mathcal{F}}{\delta\Phi_{i}},$
(1)

where the right-hand-side describes the relaxational dynamics towards the
minimum of a free energy
$\mathcal{F}=\mathcal{F}_{\text{cell}}+\mathcal{F}_{\text{int}}$, that is
composed of two parts: $\mathcal{F}_{\text{cell}}$ determines the mechanical
properties of the cell including its elasticity, compressibility and
deformability; $\mathcal{F}_{\text{int}}$ sets the strength of adhesive and
repulsive interactions between the cells. In addition to these equilibrium
contributions, the advective term $\bm{v_{i}}\cdot\nabla\Phi_{i}$, accounts
for the activity of each cell that is determined from the force balance
$\xi\bm{v_{i}}=\bm{F}_{i}^{\text{int}}+\bm{F}_{i}^{\text{active}}$, between
the active forces generated by the cells $\bm{F}_{i}^{\text{active}}$ that
include contractility [43, 44, 45, 46], substrate friction $\xi$, and the
cell-cell interaction forces $\bm{F}_{i}^{\text{int}}$ (see Supplementary
Material for the definitions of individual terms).

In order to isolate the effect of elasticity, all the physical parameters
related to the cells contractility, and cell-cell interactions, are imposed as
identical for both cell types. Simulations are initialized with a mixture of
two cell types with different elasticity positioned randomly within the domain
(Fig. 3a). In agreement with the experiments, the cells autonomously self-
segregate with time. More importantly, the cells organize themselves into a
batch of less elastic cells that are ensnared with stiffer cells at the outer
ring (Fig. 3a,b). This form of the cell segregation is consistent with the
well-established in vivo behavior in the blastocyst where the PrE-lineage
forms a layer separating the epiblast from the blastocoel [4, 38].

To quantify the cell segregation we defined the segregation index, $SI=\langle
N_{A}/N\rangle$, as the fraction of the outer cell layer that is occupied with
elastic cells, where $\langle N_{A}\rangle$ is the time-averaged number of the
elastic cells in the outer layer and $N=N_{A}+N_{B}$ is the total number of
the cells in the outer layer. Therefore, for a perfect segregation with the
more elastic cells in the outer layer $SI=1$, while $SI=1/2$ for a perfectly
mixed population, and $SI=0$ corresponds to a perfect segregation in which the
less elastic cells form the outer layer. The measurements of the segregation
index for varying elasticity difference between the cells showed that an
elasticity difference as small as 10 % is enough to trigger the cell
segregation (Fig. 3c). We further checked that the observed cell segregation
is independent of the type of the boundary and the size of the system (Fig.
3b, Fig. S5).

In addition, it is possible to pinpoint the key mechanism of the cell sorting
in the simulations. Interestingly, we found that cell sorting based on the
difference in the elasticity is only achieved in the presence of the active
stress generation by the cells (Fig. 3d). The existence of such active
stresses due to the contractile tension at the cell cortex and the myosin
activity is well documented [44, 45, 46, 12]. In our simulations we find that
as the cells continuously push and pull on their neighbors by generating
active stresses, the less elastic cells are able to deform more, exchange
neighbors at a higher rate (Fig. S6), and therefore form an inner cell
aggregate that shows a fluid-like behavior, while the more elastic cells
reside at the outer layer and do not do a considerable neighbor exchange (Fig.
S5), thus exhibiting a more solid-like behavior. Therefore, our modeling
results suggest that the cell segregation based on differential elasticity of
the cells is an inherently out of equilibrium phenomenon, which cannot be
explained by purely thermodynamic processes.

Together our results show that the observed difference in material properties,
with the PrE primed mESCs being significantly more elastic than those cells
primed for EPI differentiation, could account for the capacity of these cells
to segregate when reintroduced into blastocysts [16]. As this ESC
heterogeneity reflect the transcriptional and signaling events occurring in
the inner cell mass of the blastocyst, this suggests that differential
elasticity could function in conjunction with other mechanisms such as
differential adhesion [4, 5], to govern the second lineage segregation in
mammalian embryogenesis. This differential elasticity mechanism reinforces a
series of recent findings that showed the importance of mechanical mechanisms
such as fluid-to-solid jamming transition in determining the body axis
elongation in vertebrates [47], and differential contractility in governing
the first lineage segregation in mouse preimplantation development [6].
Moreover, the essential role of difference in material properties [48]
combined with active stress generation by the cells indicates that the non-
equilibrium physics of active materials might have played a crucial role in
the context of autonomous cell sorting during development.

## Acknowledgments

We thank William B. Hamilton and Romain Mueller for discussions and help
during the early stages of the work. The work was financially supported by the
Danish National Research Council through the StemPhys Center of Excellence,
grant nr DNRF116. A. D. acknowledges funding from the Novo Nordisk Foundation
(grant No. NNF18SA0035142 and NERD grant No. NNF21OC0068687), Villum Fonden
Grant no. 29476, and the European Union via the ERC-Starting Grant PhysCoMeT.

## References

  * Takaoka and Hamada [2012] K. Takaoka and H. Hamada, Cell fate decisions and axis determination in the early mouse embryo, Development 139, 3 (2012).
  * Wennekamp  _et al._ [2013] S. Wennekamp, S. Mesecke, F. Nedelec, and T. Hiiragi, A self-organization framework for symmetry breaking in the mammalian embryo, Nat. Rev. Mol. Cell. Biol. 14, 452 (2013).
  * Pauklin  _et al._ [2011] S. Pauklin, R. A. Pedersen, and L. Vallier, Mouse pluripotent stem cells at a glance, J. Cell Sci. 124, 3727 (2011).
  * Chazaud  _et al._ [2006] C. Chazaud, Y. Yamanaka, T. Pawson, and J. Rossant, Early lineage segregation between epiblast and primitive endoderm in mouse blastocysts through the grb2-mapk pathway, Dev. Cell 10, 615 (2006).
  * Plusa  _et al._ [2008] B. Plusa, A. Piliszek, S. Frankenberg, J. Artus, and A. K. Hadjantonakis, Distinct sequential cell behaviours direct primitive endoderm formation in the mouse blastocyst, Development 135, 3081 (2008).
  * Maitre  _et al._ [2016] J. L. Maitre, H. Turlier, R. Illukkumbura, B. Eismann, R. Niwayama, F. Nedelec, and T. Hiiragi, Asymmetric division of contractile domains couples cell positioning and fate specification, Nature 536, 344 (2016).
  * Nissen  _et al._ [2017] S. B. Nissen, M. Perera, J. M. Gonzalez, S. M. Morgani, M. H. Jensen, K. Sneppen, J. M. Brickman, and A. Trusina, Four simple rules that are sufficient to generate the mammalian blastocyst, PLoS Biol. 15, e2000737 (2017).
  * Chowdhury  _et al._ [2010] F. Chowdhury, S. Na, D. Li, Y. C. Poh, T. S. Tanaka, F. Wang, and N. Wang, Material properties of the cell dictate stress-induced spreading and differentiation in embryonic stem cells, Nat. Mater. 9, 82 (2010).
  * Vining and Mooney [2017] K. H. Vining and D. J. Mooney, Mechanical forces direct stem cell behaviour in development and regeneration, Nat. Rev. Mol. Cell Biol. 18, 728 (2017).
  * Filimonow  _et al._ [2019] K. Filimonow, N. Saiz, A. Suwinska, T. Wyszomirski, J. B. Grabarek, E. Ferretti, A. Piliszek, B. Plusa, and M. Maleszewski, No evidence of involvement of e-cadherin in cell fate specification or the segregation of epi and pre in mouse blastocysts, PLoS One 14, e0212109 (2019).
  * Balasubramaniam  _et al._ [2021] L. Balasubramaniam, A. Doostmohammadi, T. B. Saw, G. H. N. S. Narayana, R. Mueller, T. Dang, M. Thomas, S. Gupta, S. Sonam, A. S. Yap, _et al._ , Investigating the nature of active forces in tissues reveals how contractile cells can form extensile monolayers, Nat. Mater. 20, 1156 (2021).
  * Yanagida  _et al._ [2022] A. Yanagida, E. Corujo-Simon, C. K. Revell, P. Sahu, G. G. Stirparo, I. M. Aspalter, A. K. Winkel, R. Peters, H. De Belly, D. A. Cassani, _et al._ , Cell surface fluctuations regulate early embryonic lineage sorting, Cell 185, 777 (2022).
  * Riveiro and Brickman [2020] A. R. Riveiro and J. M. Brickman, From pluripotency to totipotency: an experimentalist’s guide to cellular potency, Development 147, 10.1242/dev.189845 (2020).
  * Huch and Koo [2015] M. Huch and B. K. Koo, Modeling mouse and human development using organoid cultures, Development 142, 3113 (2015).
  * Illingworth  _et al._ [2016] R. S. Illingworth, J. J. Holzenspies, F. V. Roske, W. A. Bickmore, and J. M. Brickman, Polycomb enables primitive endoderm lineage priming in embryonic stem cells, Elife 5, 10.7554/eLife.14926 (2016).
  * Canham  _et al._ [2010] M. A. Canham, A. A. Sharov, M. S. Ko, and J. M. Brickman, Functional heterogeneity of embryonic stem cells revealed through translational amplification of an early endodermal transcript, PLoS Biol. 8, e1000379 (2010).
  * Morgani  _et al._ [2013] S. M. Morgani, M. A. Canham, J. Nichols, A. A. Sharov, R. P. Migueles, M. S. Ko, and J. M. Brickman, Totipotent embryonic stem cells arise in ground-state culture conditions, Cell Rep 3, 1945 (2013).
  * Mohammed  _et al._ [2017] H. Mohammed, I. Hernando-Herraez, A. Savino, A. Scialdone, I. Macaulay, C. Mulas, T. Chandra, T. Voet, W. Dean, J. Nichols, _et al._ , Single-cell landscape of transcriptional heterogeneity and cell fate decisions during mouse early gastrulation, Cell Reports 20, 1215 (2017).
  * Pesce  _et al._ [2020] G. Pesce, P. H. Jones, O. M. Maragò, and G. Volpe, Optical tweezers: theory and practice, The European Physical Journal Plus 135, 1 (2020).
  * Maragò  _et al._ [2013] O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, Optical trapping and manipulation of nanostructures, Nature nanotechnology 8, 807 (2013).
  * Leijnse  _et al._ [2012] N. Leijnse, J.-H. Jeon, S. Loft, R. Metzler, and L. B. Oddershede, Diffusion inside living human cells, Eur. Phys. J. Special Topics 204, 75 (2012).
  * Tolic-Norrelykke  _et al._ [2004] I. M. Tolic-Norrelykke, E. L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sorensen, Anomalous diffusion in living yeast cells, Phys. Rev. Lett. 93, 078102 (2004).
  * Ghosh  _et al._ [2016] S. K. Ghosh, A. G. Cherstvy, D. S. Grebenkov, and R. Metzler, Anomalous, non-gaussian tracer diffusion in crowded two-dimensional environments, New J. Phys. 18 18 (2016).
  * Schnurr  _et al._ [1997] B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, Determining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations, Macromolecules 30, 7781 (1997).
  * Selhuber-Unkel  _et al._ [2009] C. Selhuber-Unkel, P. Yde, K. Berg-Sorensen, and L. B. Oddershede, Variety in intracellular diffusion during the cell cycle, Phys. Biol. 6, 025015 (2009).
  * Ekpenyong  _et al._ [2012] A. E. Ekpenyong, G. Whyte, K. Chalut, S. Pagliara, F. Lautenschlager, C. Fiddler, S. Paschke, U. F. Keyser, E. R. Chilvers, and J. Guck, Viscoelastic properties of differentiating blood cells are fate- and function-dependent, PLoS One 7, e45237 (2012).
  * Favre-Bulle  _et al._ [2017] I. A. Favre-Bulle, A. B. Stilgoe, H. Rubinsztein-Dunlop, and E. K. Scott, Optical trapping of otoliths drives vestibular behaviours in larval zebrafish, Nature communications 8, 1 (2017).
  * Sugden  _et al._ [2017] W. W. Sugden, R. Meissner, T. Aegerter-Wilmsen, R. Tsaryk, E. V. Leonard, J. Bussmann, M. J. Hamm, W. Herzog, Y. Jin, L. Jakobsson, _et al._ , Endoglin controls blood vessel diameter through endothelial cell shape changes in response to haemodynamic cues, Nature cell biology 19, 653 (2017).
  * Hurst  _et al._ [2021] S. Hurst, B. E. Vos, M. Brandt, and T. Betz, Intracellular softening and increased viscoelastic fluidity during division, Nature Physics 17, 1270 (2021).
  * Wessel  _et al._ [2015] A. Wessel, M. Gumalla, J. Grosshans, and C. Schmidt, The mechanical properties of early drosophila embryos measured by high-speed video microrheology, Biophysical Journal 108, 1899 (2015).
  * Gallet  _et al._ [2009] F. Gallet, D. Arcizet, P. Bohec, and A. Richert, Power spectrum of out-of-equilibrium forces in living cells: amplitude and frequency dependence, Soft Matter 5, 2947 (2009).
  * Mason and Weitz [1995] T. G. Mason and D. A. Weitz, Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids, Phys. Rev. Lett. 74, 1250 (1995).
  * Borries  _et al._ [2020] M. Borries, Y. Barooji, A. Yennek, Grapin-Botton, and L. Oddershede, Visco-elastic properties of a matrigel for organoid development as a function of polymer concentration, Frontiers in Physics 8, 579168 (2020).
  * Bird  _et al._ [1987] R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of polymeric liquids, vol. 1: Fluid mechanics, (1987).
  * Guilak  _et al._ [2000] F. Guilak, J. R. Tedrow, and R. Burgkart, Viscoelastic properties of the cell nucleus, Biochem. Biophys. Res. Commun. 269, 781 (2000).
  * Van Citters  _et al._ [2006] K. M. Van Citters, B. D. Hoffman, G. Massiera, and J. C. Crocker, The role of f-actin and myosin in epithelial cell rheology, Biophys. J. 91, 3946 (2006).
  * Chalut  _et al._ [2012] K. J. Chalut, M. Hopfler, F. Lautenschlager, L. Boyde, C. J. Chan, A. Ekpenyong, A. Martinez-Arias, and J. Guck, Chromatin decondensation and nuclear softening accompany nanog downregulation in embryonic stem cells, Biophys. J. 103, 2060 (2012).
  * Shahbazi  _et al._ [2019] M. N. Shahbazi, E. D. Siggia, and M. Zernicka-Goetz, Self-organization of stem cells into embryos: A window on early mammalian development, Science 364, 948 (2019).
  * Mueller  _et al._ [2019] R. Mueller, J. M. Yeomans, and A. Doostmohammadi, Emergence of active nematic behavior in monolayers of isotropic cells, Phys. Rev. Lett. 122, 048004 (2019).
  * Palmieri  _et al._ [2015] B. Palmieri, Y. Bresler, D. Wirtz, and M. Grant, Multiple scale model for cell migration in monolayers: Elastic mismatch between cells enhances motility, Sci. Rep. 5, 11745 (2015).
  * Wenzel and Voigt [2021] D. Wenzel and A. Voigt, Multiphase field models for collective cell migration, Physical Review E 104, 054410 (2021).
  * Jain  _et al._ [2022] H. P. Jain, D. Wenzel, and A. Voigt, Impact of contact inhibition on collective cell migration and proliferation, Physical Review E 105, 034402 (2022).
  * Prost  _et al._ [2015] J. Prost, F. Jülicher, and J.-F. Joanny, Active gel physics, Nature physics 11, 111 (2015).
  * Paluch  _et al._ [2006] E. Paluch, C. Sykes, J. Prost, and M. Bornens, Dynamic modes of the cortical actomyosin gel during cell locomotion and division, Trends in cell biology 16, 5 (2006).
  * Salbreux  _et al._ [2012] G. Salbreux, G. Charras, and E. Paluch, Actin cortex mechanics and cellular morphogenesis, Trends Cell Biol. 22, 536 (2012).
  * Chugh and Paluch [2018] P. Chugh and E. K. Paluch, The actin cortex at a glance, J. Cell Sci. 131, jcs186254 (2018).
  * Mongera  _et al._ [2018] A. Mongera, P. Rowghanian, H. J. Gustafson, E. Shelton, D. A. Kealhofer, E. K. Carn, F. Serwane, A. A. Lucio, J. Giammona, and O. Campas, A fluid-to-solid jamming transition underlies vertebrate body axis elongation, Nature 561, 401 (2018).
  * Tanaka [2022] H. Tanaka, Viscoelastic phase separation in biological cells, Communications Physics 5, 1 (2022).


select the same leaf $i$ for both $X$ and $Y$, and sample
$(\xi_{i}^{(X)\prime},\xi_{i}^{(Y)\prime})$ from a maximal coupling of the
corresponding Uniform distributions.

####  C.4.5 Move 19: rescale all missing data parameters

We propose to update all of the missingness parameters $\Xi$ via a common
$\nu$:

$\xi_{i}^{\prime}\leftarrow 1-\nu(1-\xi_{i}),\quad i\in L.$

As above, the move fails if any $\xi_{i}^{\prime}\not\in[0,1]$. As with
rescaling multiple node times by a common factor (moves 7–9), we cannot sample
from a maximal coupling here. We could attempt to couple one $\xi_{i}$ and
rescale the others accordingly but have not done so. Using the same $\nu$
variate in updating $\Xi^{(X)}$ and $\Xi^{(Y)}$ is equivalent to a common
random number coupling.

##  Appendix D Experiments

To complement the experiments with relatively weak catastrophes in Section 5.2
(strength $\kappa=0.05$, $\Gamma(1.5,5000)$ prior on the catastrophe rate
$\rho$), we now consider data generated with two larger catastrophes with
strength $\kappa=1/3$ and place a more stringent $\Gamma(1.5,10^{5})$ prior on
$\rho$. The catastrophes were placed on randomly selected branches leading
into leaf nodes before generating the data with the same death rate
$\mu=2.5\times 10^{-4}$ as M5.1. The true presence/absence state of each trait
at leaf $i\in L$ was recorded with probability
$\xi_{i}\sim\mathrm{Beta}(1,1/3)$ and marked missing otherwise. For the MCMC,
we applied a single clade constraint selected at random for each tree and set
an upper bound of $2\times 10^{3}$ on the root time, $\mu$ and $\kappa$ were
fixed at the values used to generate the data. We are not modelling lateral
transfer so only consider the number of catastrophes along each branch, the
target is the posterior distribution on the tree topology, internal node times
in $T$, the number of catastrophes on each branch $C$, and the missing data
parameters for each leaf $\Xi$.

Figure 10 displays the results of our experiments running $100$ pairs of
chains at each lag. For 8 taxa, we see in Figure 10(a) that many chains met
quickly but the rest took significantly longer, particularly for the smallest
lag $l=10^{5}$, with corresponding effects on the estimated bounds in Figure
10(b). The slow decay of ASDSF estimates on the first $5\times 10^{5}$
iterations suggests it will require many more iterations to reach the
convergence threshold of $0.01$. The coupling TV bound and ASDSF display
similar behaviour for the experiments with 12 and 16 taxa, suggesting that the
catastrophes caused fewer mixing issues in these problems.

(a) Tails of $\tau^{(l)}$ decay geometrically for each lag $l$.

(b) Estimated TV bounds eventually converge at similar rates for each lag.

(c) ASDSFs decay monotonically but fail to reach $0.01$ for the trees with 8
taxa.

Figure 10:  Diagnosing convergence on synthetic data sets with missing data,
two strong catastrophes and a stringent prior on their number.

##  Appendix E Software validation

Our algorithm is implemented in TraitLab (Nicholls, Ryder and Welch, 2013), a
Matlab toolbox for fitting Stochastic Dollo models. We validated our software
implementation through a variety of tests. For each maximal coupling of a step
of a proposal distribution, we wrote unit tests to:

  * •

check that identical pairs of states produce identical proposals;

  * •

compare the distribution of coupled draws with their marginal counterparts;

  * •

compare the proportion of identical samples with what we expect theoretically
from a maximal coupling.

For example, Figure 11 compares the proportion of matching destination
branches in a coupled SPR move with what we would expect under a maximal
coupling. We also wrote unit tests to compare the distribution of samples from
the overall couplings of the local transition kernels against their marginal
counterparts. The unit tests are included with the software. Finally, we ran
multiple pairs of chains for many iterations after they met to ensure that
chains did not decouple through an undetected bug and to verify that the
distribution of samples from coupled chains matched those from marginal
chains.

Figure 11:  Checking that our maximal coupling to sample SPR destinations
branch indices is correct. For each experiment, we sample a pair of trees with
$L$ leaves, perform housekeeping, and possibly add a random selection of
clades. For each node index $i\in V\setminus\\{r\\}$, we sampled $10^{4}$
destination index pairs $(j^{(X)},j^{(Y)})$ from a maximal coupling of
$\mathrm{U}(J_{i}^{(X)})$ and $\mathrm{U}(J_{i}^{(Y)})$, discrete Uniform
distributions on the sets of valid destination branches $J_{i}^{(X)}$ and
$J_{i}^{(Y)}$ defined in Section 4.2.1. The proportions of samples where
$j^{(X)}=j^{(Y)}$ are the _Observed_ terms in the figures. The proportion of
matching samples we would expect to see under a maximal coupling is given by
Equation 3 and are the _Expected_ terms in the figure. In each case, we see
that the observed and expected proportions match.


where $\frac{1}{p}-\frac{1}{p_{1}^{*}}=\frac{1}{nq}$. To finish with the
proof, we can use a bi-parameter version of the truncation method to jump to
the strong bound (see Appendix A.2). ∎

##  7\. Proofs part III: the local Biparameter Fractional Poincaré-Sobolev
inequality

In this section we will provide a proof of Theorem 2.20. We use the main
argument from Section 6 for proving Theorem 2.16 with a different functional
$a(R)$ given by Theorem 2.18.

###### Proof of Theorem 2.20.

From Theorem 2.18 with $p_{1}=p_{2}=p$ we have

$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}|f-f_{R}|\leq
A+B$

where

$A\leq
c_{n_{1}}(1-\delta)^{\frac{1}{p}}\ell(I_{1})^{\delta}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|^{p}}{|x_{1}-y_{1}|^{n_{1}+p\delta}}\,dy_{1}dx_{1}\
dx_{2}\right)^{\frac{1}{p}}$

and

$B\leq
c_{n_{2}}\,(1-\delta)^{\frac{1}{p}}\ell(I_{2})^{\delta}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{R}\int_{I_{2}}\frac{|f(y_{1},x_{2})-f(y_{1},y_{2})|^{p}}{|x_{2}-y_{2}|^{n_{2}+p\delta}}\
dy_{1}dx_{2}dy_{2}.\right)^{\frac{1}{p}}$

Now, recalling the definition of $A_{1,\mathfrak{R}}$ given in (6.3):

$\frac{1}{|R|}\int_{R}w(x)\,dx\leq[w]_{A_{1,\mathfrak{R}}}\,\inf_{R}w\qquad
R\in\mathfrak{R}$

we have,

$A\leq
c_{1}\,[w]_{A_{1,\mathfrak{R}}}^{\frac{1}{p}}\,(1-\delta)^{\frac{1}{p}}\ell(I_{1})^{\delta}\left(\frac{1}{w(R)}\int_{R}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|^{p}}{|x_{1}-y_{1}|^{n_{1}+p\delta}}\,w(x_{1},x_{2})dx_{1}dx_{2}\
dy_{1}\right)^{\frac{1}{p}}.$

Similarly we have for $B$

$B\leq
c_{2}\,[w]_{A_{1,\mathfrak{R}}}^{\frac{1}{p}}\,(1-\delta)^{\frac{1}{p}}\ell(I_{2})^{\delta}\left(\frac{1}{w(R)}\int_{R}\int_{I_{2}}\frac{|f(y_{1},x_{2})-f(y_{1},y_{2})|^{p}}{|x_{2}-y_{2}|^{n_{2}+p\delta}}\,w(y_{1},y_{2})\,dy_{1}dy_{2}\
dx_{2}\right)^{\frac{1}{p}}$

Let

$a_{1}(R)=c_{1}\,[w]_{A_{1,\mathfrak{R}}}^{\frac{1}{p}}\,(1-\delta)^{\frac{1}{p}}\ell(I_{1})^{\delta}\left(\frac{1}{w(R)}\int_{R}\int_{I_{1}}\frac{|f(x_{1},x_{2})-f(y_{1},x_{2})|^{p}}{|x_{1}-y_{1}|^{n_{1}+p\delta}}\,w(x_{1},x_{2})dx_{1}dx_{2}\
dy_{1}\right)^{\frac{1}{p}}.$

and

$a_{2}(R)=c_{2}\,[w]_{A_{1,\mathfrak{R}}}^{\frac{1}{p}}\,(1-\delta)^{\frac{1}{p}}\ell(I_{2})^{\delta}\left(\frac{1}{w(R)}\int_{R}\int_{I_{2}}\frac{|f(y_{1},x_{2})-f(y_{1},y_{2})|^{p}}{|x_{2}-y_{2}|^{n_{2}+p\delta}}\,w(y_{1},y_{2})\,dy_{1}dy_{2}\
dx_{2}\right)^{\frac{1}{p}}$

Observe that each functional $a_{j}$ is of the form (6.6) since each of the
inner integrand of the functional, $A(R,x)$ is increasing in $R$,

$R_{1}=I_{1}\times I_{2}\subseteq R_{2}=J_{1}\times J_{2}\quad\iff\quad
I_{1}\subseteq J_{1}\text{ and }I_{2}\subseteq J_{2}$

and we can apply Lemma 6.2. Then if $M>1$, $a_{j}\in
SD^{s}_{p_{w}^{*},\mathfrak{R}}(w)$ with $s=\frac{nM^{\prime}}{\delta}$ and
$\|a\|_{SD^{s}_{p_{w}^{*},\mathfrak{R}}(w)}\leq[w]_{A_{q,\mathcal{R}}}^{\frac{\delta}{nqM}}$.

Since the functionals from (6.6) satisfy the $SD_{p_{w}^{*},\mathfrak{R}}^{s}$
condition given in Lemma 6.2, (3), namely

$\frac{1}{p}-\frac{1}{p_{w}^{*}}=\frac{\delta}{n}\frac{1}{M}.$

$\left(\sum_{i}a_{j}(R_{i})^{p^{*}_{w}}\,\frac{w(R_{i})}{w(R)}\right)^{\frac{1}{p^{*}_{w}}}\leq[w]_{A_{1,\mathfrak{R}}}^{\frac{\delta}{nM}}\,\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta}{nM^{\prime}}}a(R)\qquad
j=1,2$

and hence the corresponding result holds for

$a(R):=a_{1}(R)+a_{2}(R),$

more precisily

$\left(\sum_{i}(a(R_{i}))^{p^{*}_{w}}\,\frac{w(R_{i})}{w(R)}\right)^{\frac{1}{p^{*}_{w}}}\leq
2[w]_{A_{1,\mathfrak{R}}}^{\frac{\delta}{nM}}\,\left(\frac{|\bigcup_{i}R_{i}|}{|R|}\right)^{\frac{\delta}{nM^{\prime}}}a(R)\qquad
j=1,2$

Now we are in a position to finish the proof exactly as of Theorem 2.16. We
omit the details.

∎

##  Appendix A The truncation method

We include here for completeness the different truncation arguments that we
used in our results.

###  A.1. The classical weak implies strong for the gradient

We include here the classical truncation argument for Lipschitz functions
related to the functional defined by the integral of the gradient. The general
idea of truncation arguments in the context of Poincaré-Sobolev inequalities
is well known, classical references are [M85] and [BCLSC95].

######  Lemma A.1.

Let $g:\mathbb{R}^{n}\to{\mathbb{R}}$ be any nonnegative Lipschitz function.
Suppose that for the pair $1\leq q<p$ there is a weak type estimate for a pair
of measures $\nu,\mu$ of the form:

$\|g\|_{L^{p,\infty}_{\mu}}\lesssim\left(\int_{{\mathbb{R}}^{n}}|\nabla
g|^{q}\,d\nu\right)^{\frac{1}{q}}.$

Then the strong estimate also holds, namely

$\|g\|_{L^{p}_{\mu}}\lesssim\left(\int_{{\mathbb{R}}^{n}}|\nabla
g|^{q}\,d\nu\right)^{\frac{1}{q}}.$

###### Proof.

Consider the usual truncation of a non negative function $g$ at level $2^{k}$
given by $T_{k}(g)$ defined by:

$T_{k}g(x):=\left\\{\begin{array}[]{cc}0&\text{ if }g(x)\leq 2^{k}\\\
g(x)-2^{k}&\text{ if }2^{k}<g(x)<2^{k+1}\\\ 2^{k}&\text{ if }g(x)\geq
2^{k+1}.\\\ \end{array}\right.$

Also define for each $k\in\mathbb{Z}$ the set
$G_{k}:=\\{x\in\mathbb{R}^{n}:2^{k}<g(x)\leq 2^{k+1}\\}$. We have that, for
all $x\in G_{k+1}$, $T_{k}g(x)=2^{k}$ and $\text{sop}\nabla(T_{k}g)\subset
G_{k}$. We proceed as follows:

$\displaystyle\int_{\mathbb{R}^{n}}g^{p}(x)d\mu$ $\displaystyle\lesssim$
$\displaystyle\sum_{k=-\infty}^{k=\infty}2^{kp}\mu(G_{k+1})$
$\displaystyle\lesssim$
$\displaystyle\sum_{k=-\infty}^{k=\infty}2^{(k-1)p}\mu(T_{k}g(x)>2^{k-1})$
$\displaystyle\lesssim$
$\displaystyle\sum_{k=-\infty}^{k=\infty}\left(2^{k-1}\mu(T_{k}g(x)>2^{k-1})^{\frac{1}{p}}\right)^{p}.$

Since $T_{k}g$ is still a Lipschitz function, we can use the hypothesis and
get

$\displaystyle\int_{\mathbb{R}^{n}}g^{p}(x)d\mu$ $\displaystyle\lesssim$
$\displaystyle\sum_{k=-\infty}^{k=\infty}\left(\int_{{\mathbb{R}}^{n}}|\nabla
T_{k}g|^{q}\,d\nu\right)^{\frac{p}{q}}$ $\displaystyle\lesssim$
$\displaystyle\sum_{k=-\infty}^{k=\infty}\left(\int_{G_{k}}|\nabla
T_{k}g|^{q}\,d\nu\right)^{\frac{p}{q}}$ $\displaystyle\lesssim$
$\displaystyle\sum_{k=-\infty}^{k=\infty}\left(\int_{G_{k}}|\nabla
g|^{q}\,d\nu\right)^{\frac{p}{q}}$ $\displaystyle\lesssim$
$\displaystyle\left(\int_{\mathbb{R}^{n}}|\nabla
g|^{q}\,d\nu\right)^{\frac{p}{q}}$

using that $\frac{p}{q}>1$ and the disjointness of the family $\\{G_{k}\\}$. ∎

###  A.2. Truncation in the bi-parameter setting

######  Lemma A.2.

Let $g:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\to{\mathbb{R}}$ be any
nonnegative Lipschitz function. Suppose that for $p\geq 1$ there is a weak
$(q,p)$-type estimate for a pair of measures $\nu,\mu$ of the form:

$\|g\|_{L^{p,\infty}_{\mu}}\lesssim\left(\int_{{\mathbb{R}}^{n_{1}}\times{\mathbb{R}}^{n_{2}}}(|\nabla_{1}g|^{q}d\nu\right)^{1/q}+\left(\int_{{\mathbb{R}}^{n_{1}}\times{\mathbb{R}}^{n_{2}}}(|\nabla_{2}g|^{q}d\nu\right)^{1/q}.$

Then the strong estimate also holds, namely

$\|g\|_{L^{p}_{\mu}}\lesssim\left(\int_{{\mathbb{R}}^{n_{1}}\times{\mathbb{R}}^{n_{2}}}(|\nabla_{1}g|^{q}d\nu\right)^{1/q}+\left(\int_{{\mathbb{R}}^{n_{1}}\times{\mathbb{R}}^{n_{2}}}(|\nabla_{2}g|^{q}d\nu\right)^{1/q}.$

###### Proof.

This case is as easy as Lemma A.1, since the class of Lipschitz functions
enjoys the truncation property. The same proof works here as well. ∎

##  Appendix B The proof of Lemma 5.3

We include here the proof of Lemma 5.3, namely estimate (5.4). We recall here
that estimate:

(B.1)
$\ell(Q)^{\delta}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|f(x)-f(y)|}{|x-y|^{n+\delta}}dydx\lesssim_{n}\frac{1}{\delta(1-\delta)}\ell(Q)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\nabla
f|.$

###### Proof of Lemma 5.3.

We start by using the FTC to obtain a representation formula as follows

Using the FTC, for every $x,y\in Q$,

$f(y)-f(x)=\int_{0}^{1}\nabla f(x+t(y-x))\cdot(y-x)dt.$

Then, for a fixed $x\in Q$, the inner integral in (B.1) can be written as

$\displaystyle\int_{Q}\frac{|f(x)-f(y)|}{|x-y|^{n+\delta}}dydx$
$\displaystyle\leq$ $\displaystyle\int_{Q}\int_{0}^{1}\frac{|\nabla
f(x+t(y-x))|}{|x-y|^{n+\delta-1}}dtdydx$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{Q\cap B(x,\sqrt{n}\ell(Q))}\frac{|\nabla
f(x+t(y-x))|}{|x-y|^{n-(1-\delta)}}dydtdx$ $\displaystyle=$ $\displaystyle I$

since $Q\subset B(x,\sqrt{n}\ell(Q))$ for any $x\in Q$. Now, we change
variables putting $z=x+t(y-x)=(1-t)x+ty$. Then one has $|x-y|=|z-x|/t$ and the
change of variables theorem yields

$\displaystyle I$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{((1-t)x+tQ)\cap
B(x,\sqrt{n}t\ell(Q))}\frac{|\nabla
f(z)|}{|z-x|^{n-(1-\delta)}}\frac{t^{n-(1-\delta)}}{t^{n}}dzdtdx$
$\displaystyle\leq$ $\displaystyle\int_{0}^{1}\int_{Q\cap
B(x,\sqrt{n}t\ell(Q))}\frac{|\nabla
f(z)|}{|z-x|^{n-(1-\delta)}}t^{-(1-\delta)}dzdtdx$ $\displaystyle\leq$
$\displaystyle\int_{Q}\int_{\frac{|z-x|}{\sqrt{n}\ell(Q)}}^{1}\frac{dt}{t^{1-\delta}}\frac{|\nabla
f(z)|}{|z-x|^{n-(1-\delta)}}dzdx$

by Fubini again. Going back to the beginning and putting all together we
obtain

$\frac{\ell(Q)^{\delta}}{\delta}\,\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|\nabla
f(z)|}{|z-x|^{n-(1-\delta)}}dzdx=\frac{\ell(Q)^{\delta}}{\delta}\,\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\nabla
f(z)|\int_{Q}\frac{dx}{|z-x|^{n-(1-\delta)}}dz.$

Now we use that, for any Lebesgue measurable set $\Omega$ and $0<\alpha<n$, we
have the estimate

$\int_{\Omega}\frac{dx}{|z-x|^{n-\alpha}}\leq
v_{n}^{-\alpha/n}\alpha^{-1}|\Omega|^{\alpha/n},\qquad\text{for all }z,$

where $v_{n}$ is the volume of the unit ball of $\mathbb{R}^{n}$. We finally
obtain that

$\displaystyle\ell(Q)^{\delta}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}\int_{Q}\frac{|f(x)-f(y)|}{|x-y|^{n+\delta}}dydx$
$\displaystyle\leq$ $\displaystyle
c_{n}\ell(Q)^{\delta}\frac{1}{\delta(1-\delta)}\,|Q|^{\frac{1-\delta}{n}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\nabla
f(z)|$ $\displaystyle=$
$\displaystyle\frac{c_{n}}{\delta(1-\delta)}\,\ell(Q)\,\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{Q}|\nabla
f(z)|.$

∎

## References

  * [AD04] G. Acosta and R. Durán, _An optimal Poincaré inequality in $L^{1}$ for convex domains_, Proc. Amer. Math. Soc. 132 (2004), no. 1, 195–202.
  * [AH96] D. R. Adams, L. I. Hedberg, Function spaces and potential theory. Die Grundlehren der Mathematischen Wissenschaften 314, Springer-Verlag, Berlin, 1996. xii+366 pp.
  * [BCLSC95] D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44, (1995), no. 4, 1032–1074.
  * [BBM02] J. Bourgain, H. Brezis, and P. Mironescu, _Limiting embedding theorems for $w^{s;p}$ when $s\uparrow 1$ and applications_, J. Anal. Math. 87 (2002), 77–101.
  * [BVY21] H. Brezis, J.Van Schaftingen and P.-L. Yung, _A surprising formula for Sobolev norms and related topics,_ Proc. Natl. Acad. Sci. 118 (2021) no. 8.
  * [CP21] J. Canto and C. Perez, _Extensions of the John - Nirenberg theorem and applications_ , Proc. Amer. Math. Soc. 149 (2021), 1507-1525.
  * [CPR] J. Canto, C. Perez, and E. Rela, _Minimal conditions for BMO_ , arXiv: https://arxiv.org/abs/2007.10104.
  * [CDM19] E. Cejas, I. Drelichman and J. Martínez-Perales, Improved fractional Poincaré type inequalities on John domains, Ark. Mat. 57.2 (2019), pp. 285–316.
  * [Chu95] S. Chua, _Weighted sobolev inequalities of mixed norm_ , Real Anal. Ex. 21 (1995), no. 2, 555–571.
  * [Chu06] S. Chua, _Extension theorems on weighted Sobolev spaces and some applications_ , Canad. J. Math. 58 (2006), no. 3, 492–528.
  * [NPV12] E. Di Nezza, G.Palatucci, and E. Valdinoci, _Hitchhiker’s guide to the fractional Sobolev spaces_ , Bull. Sci. Math. 136 (2012), no 5, 521-573.
  * [DM10] O. Domínguez and M. Milman, _New Brezis-Van Schaftingen-Yung Sobolev type inequalities connected with maximal inequalities and one parameter families of operators,_ arXiv:2010.15873v1.
  * [DD] I. Drelichman and R. G. Durán, _The Bourgain-Brézis-Mironescu formula in arbitrary bounded domains,_ arXiv:2012.14505v2.
  * [DIV16] B. Dyda, L. Ihnatsyeva, L. and A. Vähäkangas, On improved fractional Sobolev-Poincaré inequalities, Ark. Mat. 54.2 (2016), pp. 437-454.
  * [FPW98] B. Franchi, C. Pérez, and R. Wheeden, _Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type_ , J. Funct. Anal. 153 (1998), no. 1, 108–146.
  * [Fuj78] N. Fujii, _Weighted bounded mean oscillation and singular integrals_ , Math. Japon. 22 (1977/78), no. 5, 529–534.
  * [Ha01] P. Hajlasz, Sobolev inequalities, truncation method, and John domains, Papers on analysis, V. 83 109-126, Rep. Univ. Jyväskylä Dep. Math. Stat., 83, Univ. Jyväskylä, Jyväskylä, 2001.
  * [HaK95] P. Hajlasz and P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris, 320, (1995), 1211–1215.
  * [HKST15] J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson, _Sobolev Spaces on Metric Measure Spaces An Approach Based on Upper Gradients_ , Cambridge University Press, New Mathematical Monographs (27) ,2015.
  * [HV13] R. Hurri-Syrjänen, A. Vähäkangas. On fractional Poincaré inequalities,. J. Anal. Math. 120 (2013) , 85–104.
  * [HMPV] R. Hurri-Syrjänen, J. Martínez , C. Pérez and A. Vähäkangas. On local fractional Poincaré inequalities with weights, in preparation
  * [HP13] T. Hytönen and C. Pérez, _Sharp weighted bounds involving ${A}_{\infty}$_, Anal. PDE 6 (2013), no. 4, 777–818.
  * [HPR12] T. Hytönen, C. Pérez, and E. Rela, _Sharp Reverse Hölder property for $A_{\infty}$ weights on spaces of homogeneous type_, J. Funct. Anal. 263 (2012), no. 12, 3883–3899.
  * [Kar02] G. Karagulyan, _Exponential estimates for the Calderón-Zygmund operator and related problems of Fourier series_ , Mat. Zametki 71 (2002), no. 3, 398–411.
  * [KLV] J. Kinnunen, J. Lehrbäck and A. Vähäkangas _Maximal function methods for Sobolev spaces_ , Mathematical Surveys and Monographs, Amer. Math. Soc., to be published.
  * [LP05] A. Lerner and C. Pérez, _Self-improving properties of generalized Poincaré type inequalities through rearrangements_ , Math. Scand. 97 (2005), no. 2, 217–234.
  * [LSSVZ15] A. Logunov, L. Slavin, D. Stolyarov, V. Vasyunin, and P. Zatitskiy, _Weak integral conditions for BMO_ , Proc. Amer. Math. Soc. 143 (2015), no. 7, 2913–2926.
  * [LP02] G. Lu and C. Pérez, _$L^{1}\to L^{q}$ Poincaré inequalities for $0<q<1$ imply representation formulas_, Acta Math. Sin. (Engl. Ser.) 18 (2002), no. 1, 1–20.
  * [LW98] G. Lu and R. Wheeden, _Poincaré inequalities, isoperimetric estimates, and representation formulas on product spaces_ , Indiana Univ. Math. J. 47 (1998), no. 1, 123–151.
  * [LPR17] T. Luque, C. Pérez, and E. Rela, _Reverse Hölder property for strong weights and general measures_ , J. Geom. Anal., 27/1 (2017), 162–182.
  * [MP98] P. MacManus and C. Pérez, _Generalized Poincaré inequalities: sharp self-improving properties_ , Internat. Math. Res. Notices (1998), no. 2, 101–116.
  * [M85] V. Maz’ya, Sobolev spaces. Transl. from the Russian by T. O. Shaposhnikova, Berlin etc.: Springer-Verlag. xix, 486 p. (1985)
  * [MS02] V. Maz’ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195, (2002) 230-238.
  * [M05] M. Milman, _Notes on limits of Sobolev spaces and the continuity of interpolation scales,_ Trans. Amer. Math. Soc. 357 (2005), 3425-3442.
  * [OP02] J. Orobitg and C. Pérez, _$A_{p}$ weights for nondoubling measures in ${\mathbb{R}}^{n}$ and applications_, Trans. Amer. Math. Soc. 354 (2002), no. 5, 2013–2033 (electronic).
  * [PR18] I. Parissis and E. Rela, _Asymptotically sharp reverse hoelder inequalities for flat muckenhoupt weights_ , Indiana Univ. Math. J. 67 (2018), 2363–2391.
  * [PR19] C. Pérez and E. Rela, _Degenerate Poincaré-Sobolev inequalities_ , Trans. Amer. Math. Soc. 372 (2019), no. 9, 6087–6133.
  * [SC92] L. Saloff–Coste, A note on Poincaré, Sobolev and Harnack inequalities, International Mathematics Research Notices 2 (1992), 27–38.
  * [ST93] X. Shi and A. Torchinsky, _Poincaré and Sobolev inequalities in product spaces_ , Proc. Amer. Math. Soc. 118 (1993), no. 4, 1117–1124.
  * [T00] B. O. Turesson: Nonlinear potential theory and weighted Sobolev spaces. Lecture Notes Math. 1736. Springer-Verlag, Berlin, 2000.