# Nucleation of cadherin clusters on cell-cell interfaces

Neil Ibata and Eugene M. Terentjev<EMAIL_ADDRESS>
###### Abstract

Cadherins mediate cell-cell adhesion and help the cell determine its shape and
function. Here we study collective cadherin organization and interactions
within cell-cell contact areas, and find the cadherin density at which a ‘gas-
liquid’ phase transition occurs, when cadherin monomers begin to aggregate
into dense clusters. We use a 2D lattice model of a cell-cell contact area,
and coarse-grain to the continuous number density of cadherin to map the model
onto the Cahn-Hilliard coarsening theory. This predicts the density required
for nucleation, the characteristic length scale of the process, and the number
density of clusters. The analytical predictions of the model are in good
agreement with experimental observations of cadherin clustering in epithelial
tissues.

## Introduction

Many eukaryotic cells use membrane-bound integrin adhesion clusters [1] to
tether themselves to the extra-cellular matrix surrounding them (ECM).
Similarly, these cells use membrane-bound cadherin adhesion clusters [2, 3, 4]
to bind to their neighbouring cells directly. Adhesion molecules mediate
mechanical signalling between the cell and its exterior by participating in
important intracellular signalling pathways [5, 6, 7, 8]. Clusters of adhesion
molecules also help determine the structure of the cell; these shape changes
are essential if the cell is to topologically fit into a tissue (e.g. in
dividing epithelia [9]), to change its function (fibroblasts differentiating
into myofibroblasts when placed on stiff media [10, 11]), or to move (during
wound healing [12, 13] or cancer metastasis [14, 15]). In order to understand
why any of these processes occur, we must first understand why there are
clusters of adhesion molecules at all – in physical terms, how their
nucleation from a uniform distribution of sensors occurs.

Density-dependent nucleation is ubiquitous in soft matter. Changes in the
concentration of attractively coupled molecules can help form large-scale
symmetry-breaking structures. Computational studies have investigated the
clustering of both integrin [16, 17] and cadherin [18, 19]. The classical
theory of aggregation on fluctuating membranes [20, 21, 22] explains the
unstable growth of nuclei into large-scale receptor domains (evident in low-
resolution experiments), notably including the case of large cadherin domains
on vesicles [23]. These effects are driven by the weak long-range forces
mediated by membrane fluctuations, and cannot account for the stability of
small nanometer-scale clusters of the kind recently identified in high-
resolution cadherin imaging [24, 25]. We recently analytically investigated
the nucleation of integrins in the high-concentration limit on the edge of a
spreading cell [26]. Here, we extend this approach to the nucleation of such
small punctate cadherin clusters on a generic cell-cell contact surface,
stabilized by the strong short-range bonding between monomers.

We will first review background literature to examine the parallels between
the aggregation of the types of adhesion molecules. Next, we build a lattice
gas model to obtain the Ginzburg-Landau free energy for fluctuations in the
density of cadherin molecules on the realistic 2D contact plane between cells.
We consider whether our model correctly predicts the size and spacing of
punctate adherens junctions seen in experiment.

Integrin/Cadherin analogy. The cell needs to develop adhesion clusters in
order to correctly remodel its shape in response to external substrates and
mechanical cues [27, 28]. The large clusters visible under microscope (called
focal adhesions for integrins [10, 29], and zonula adherens for cadherin [30,
31]) are the end-product of many smaller clusters aggregating over minutes or
hours.

Figure 1: Left panel: E-cadherin (‘E’ standing for epithelial) aggregate into
punctate clusters with a characteristic size $\lesssim 50$ nm and separation
$\gtrsim 100$ nm. Reproduced with permission from Wu et al. [24]. Right panel:
Aggregation of cadherins (1) and integrins (2). a) Single adhesion molecules
sit in the cell membrane, inactive. b1) Catenins begin to aggregate around the
transmembrane cadherins, which form transmembrane trans-bonds, and link to the
cytoskeleton [3, 32]: an individual adhesion complex forms. b2) Integrins are
activated by pulling when an individual adhesion complex assembles [33]. c)
Adhesion units aggregate laterally via cadherin cis-bonds (1) or auxiliary
proteins for integrin (2), and form clusters.

In order to be stabilized, both integrins [34] and cadherins [35] rely on
‘catch bonds’ [36], which strengthen under load. These bonds can help activate
both molecules and are often preceded by the formation of a larger cytoplasmic
protein complex which links an individual adhesion molecule to the
cytoskeleton (see Fig. 1). Integrin uses talin and vinculin to bind to F-actin
[37, 38], while cadherin primarily relies on catenins [39, 40]. After the
protein complex is fully assembled, the acto-myosin cytoskeleton exerts
pulling force on the adhesion molecule, strengthening the catch bond.
Individual complexes can then aggregate into growing clusters, and link the
cytoskeleton with the outside of the cell over a larger area, spreading the
force applied by the actomyosin cortex [41, 42]. Provided that adhesion
molecules cluster together in sufficient numbers to withstand the load, the
cytoskeleton can develop increasingly large pulling forces and distort the
shape of the cell [43].

Cadherins differ most notably from integrins in their ability to multimerise.
Cadherins can form two types of bonds with other cadherins: trans bonds that
link cadherins from two different cells [44, 45], which effectively mimic the
integrin binding to extra-cellular matrix, and cis bonds that link
neighbouring cadherins within the same cell, in the plane of the membrane.
While integrins and cadherins might develop a very similar force-chain with
the cytoskeleton in the direction normal to the cell membrane, their
interactions within the plane of the cell membrane are fundamentally
different. Whereas integrins appear to use secondary molecules to link to each
other (possibly $\alpha$-actinin [46, 47, 48, 49]), cadherins bind directly to
each other [50].

The distance between cadherin monomers in a loosely packed cadherin lattice is
ca.$7$ nm [51, 52, 53], but can be as small as $3$ nm in very tightly packed
junctions [54]; both of these figures are much smaller than the separation
than between neighbouring integrins: ca.$30$ nm [55]. The cis bond strength
can be estimated by extrapolating from measurements of cadherin trans bond
strength and the dissociation rates of these bonds, as well as numerical
simulations of cadherin clustering. Note that it is much faster than the
dissociation of cadherins in well-developed adhesion junctions with many such
bonds established [56].

Over the last ten years, the classical view has shifted from the idea that
cis-dimerisation preceded trans-dimerisation [57]. The discovery of an
intermediate crossed-dimer, called the X-dimer, which precedes a more stable
swapped trans-dimerisation suggests that cis-dimerisation is not necessary for
cadherins to form stable cross-membrane bonds [50, 51]. More recent work [58]
suggests that cadherin cross-membrane dimers form before cis clustering
occurs. This can be further quantified by examining recent work on the
development of large adherent regions in giant unilamellar vesicles (GUVs). In
particular, Fenz et al. [23] showed that cadherin trans-bonding, a
characteristic of the extended adhesive region, spread from nucleation sites
if membrane fluctuations were sufficiently small. In Supplementary Part A, we
show that the size of membrane fluctuations within an adhesive region in a
cell is much smaller than that required for adhesive regions to separate into
separate domains in this model, based on the data from [59, 60]. This means
that within an adhesive region, a large portion of cadherins are indeed within
trans-bonded dimers (with on-rates $\ll 1$ s, given an intrinsic lifetime of
$0.63$ s [35]). Lateral interactions between cadherin trans-bonded pairs are
small in this model (a few $k_{B}T$ at most), relying on membrane fluctuations
to slowly and randomly bring cadherin pairs together. In contrast, small
crystalline clusters of cadherin have been recently found [24, 25] (see Fig.
1), and these disappear if the cis-abolishing V81D/V175D mutation is
introduced [51]. This means that while cadherin trans-interaction mediated by
membrane fluctuations does lead to the development of the larger adhesive
area, it is insufficient to help develop individual punctate cadherin
adhesions.

Cadherin trans-bond strength has been suggested to be in the range
$9$-$13k_{B}T$, substantially greater than the $J\leq 7k_{B}T$ suggested for
cis-bond [24]. The on-rates of trans bonds should therefore be $10$-$100$
times greater than those of cis bonds, and we expect, as in much of the
literature, for trans bonds to form before cis bonds. The problem then no
longer needs to be resolved separately in both cells; rather, we need only
look at the cis (in plane) clustering of the cadherin trans-dimers on the
adhesion surface. Recent computational work by Yu et al. [61] (see their
Figure 3.B) gives yet more credence to the idea that at both trans and cis-
bonds are required to form punctate adherens junctions with substantial
numbers of cadherins of the kind seen by Wu et al. [50], and that there is an
optimum cis-bond strength for cluster formation of $3$-$8k_{B}T$. We take
$5k_{B}T$ as a compromise value, given previous research suggesting that cis
bond strength is smaller than trans [50].

The effective binding strength of the each cadherin trans-dimer unit to its
neighbours is then twice as strong as the monomer value: $2J\approx 10k_{B}T$.
The remainder of this work will assume that all cadherins that will form
clusters have trans-dimerised, and that the other cadherins do not impede the
pair interaction.

To simplify terminology and to make the comparison with the aggregation of
integrin complexes more apparent, we will henceforth use the term “adhesion
complex” to denote a transmembrane cadherin trans-dimer, bound to catenin and
to the actin cytoskeleton insoide each cell (Fig. 1). We seek to examine the
initial nucleation of cadherin clusters revealed in super-resolution
microscopy [24]. This occurs before the aggregation coarsens into large
adherens junctions [62].

## The model and methods

We recently showed [26] how attractively coupled integrin adhesion complexes
modelled on a 1D lattice undergo a density-dependent phase transition if their
initially high concentration decreases past a critical value as the lattice
length increases while their number is held constant. This was to model the
development of nascent focal adhesions during cell spreading and correctly
predicted the number of adhesion clusters.

The analogy between integrin and cadherin bonding suggests that a lattice gas
model might be applicable to the nucleation of cadherin clusters. However, the
cell-cell interface is fundamentally 2D (as opposed to integrins concentrated
on the rim of the contact area), and we will find that the transition here
happens as density increases, more in line with a gas-liquid condensation
(again, different from the integrin transition). Accordingly, we model the
contact plane between two cells as a 2D lattice with Neumann boundary
conditions, with a total number of lattice sites $A$, and total area
$\Sigma=a^{2}A$, with $a$ the lattice spacing or the distance between cadherin
trans dimers. Changes in the density of cadherins are introduced by the
adiabatic change the area of the lattice.

The Hamiltonian of the attractively coupled cadherin trans-bonded pairs is an
adaptation of the Ising model:

$H=-2J\Sigma_{\langle ij\rangle}\eta_{i}\eta_{j}\ ,$ (1)

where the sum is over nearest-neighbor pairs. The variable $\eta$ keeps track
of the occupation number of each site:

$\displaystyle\eta_{i}=\begin{cases}&1\ \ \ \ {\text{adhesion complex
present}}\\\ &0\ \ \ \ {\text{adhesion complex absent}}\end{cases}\ \ \ \ .$
(2)

The binding energy $2J$ between two trans-dimerised cadherin adhesion
complexes is double the monomer binding energy $J$. We show in the
Supplementary Materials that a cytoskeletal pulling force, contributiong to
the Hamiltonian with a linear term $-\Sigma_{i}h\eta_{i}$, does not effect the
size or distance between clusters, or the concentration at which aggregation
begins.

In order to examine the spatial clustering of cadherins, we need to derive the
Ginzburg-Landau action of the density distribution of cadherin units. In
addition, the diffusion time over the area of membrane that separates the
punctate adhesions (50-100 nm) is ca.$1$ s, similar to the growth time of the
cluster (see below). The onset of aggregation is faster still, so the number
of cadherins within the area which collapses into a single adhesion (the local
$N/A$ value) should not change substantially at this crucial point.

Over a sufficiently short time interval, the contact area $A$ can be assumed
constant, and we need not worry about a changing expression for the partition
function. The single-molecule partition function for the lattice gas model
with Hamiltonian (1) is:

$Z_{i}=\Sigma_{\eta_{i}=\\{0,1\\}}e^{-\beta[-2J\Sigma_{\langle
j\rangle}\eta_{j}]\cdot\eta_{i}}\ ,$ (3)

where the sum runs over all of the $A$ sites in the contact area. The full
partition function is the product of all $Z_{i}$, subject to the constraint of
the constant total number of individual cadherin units,
$N=\Sigma_{i}\eta_{i}$. The Supplementary Part B gives the calculation of this
partition function using the auxilliary fields method. There, we introduce a
site-specific variable $\rho_{i}$, whose expectation value is the average
occupation of a site $\langle\eta_{i}\rangle$, and later transform this into a
continuous density $\rho(s)$ which depends on a the position $s$ in the
contact area.

The calculation of
$Z_{\text{tot}}=\delta(\Sigma_{i}\eta_{i}-N)\Pi_{i=1}^{A}Z_{i}$ is exact, but
rather unwieldy. We need to make two strong assumptions if we want a
manageable form for the effective action $S[\rho]$. First (discussed in
Supplementary Part B), the concentration of sensors is assumed low, so that
the probability of a single site being in the ‘empty’ state is much greater
than that of being in the ‘filled’ state. Second (Supplementary Part C), we
look at the nucleation of clusters, where non-uniformity amplitude is small,
so we can work with the series expansion of $S$ in terms of density
fluctuations $\phi=\rho-N/A$.

Note that first order terms in the new variable $\phi$ average to zero, and
only result in a constant shift in the action. This is why the strength of the
cytoskeletal pulling force encapsulated in the field term $h$ does not change
the kinetics of cadherin punctate adhesion aggregation.

In Supplementary Part D, we obtain the action $S[\rho_{i}]$, transform it into
Fourier space, make it continuous, and finally transform back into real space
(the last operation generating the spatial gradients). It has recognizable
features of the Ginzburg-Landau theory, where we retain cubic and quartic
terms in the order parameter expansion:

$\displaystyle S_{\Lambda_{0}}[\mathbf{\phi}]=\int
d\mathbf{s}\Big{[}\frac{r_{0}}{2}\phi^{2}(\mathbf{s})+\frac{c_{0}}{2}[\nabla\phi(\mathbf{s})]^{2}+\frac{t_{1}}{3!}\phi^{3}(\mathbf{s})+\frac{u_{1}}{4!}\phi^{4}(\mathbf{s})+\frac{u_{2}}{4!}\phi^{2}(\mathbf{s})\Big{(}\int
d\mathbf{s}^{\prime}\phi^{2}(\mathbf{s}^{\prime})\Big{)}\Big{]}\ ,$ (4)

where all lengths are scaled by the size of the individual sensor $a$, and the
coefficients are listed in Supplementary Part D. Specifically, the two
quadratic-order coefficients take the form, Eqn. (D.9):

$r_{0}=8\beta J(1-8g_{2}\beta J)\ ;\ \ \ c_{0}=32g_{2}\beta^{2}J^{2}-2\beta J\
,$ (5)

with $g_{2}={N}/{A}-{N^{2}}/{(A+N)^{2}}$, from Eqn. (C.5).

The control parameter (replacing the temperature in the classical theory of
phase transitions) is the ratio $N/A$, which starts near 0 for an initially
large contact plane, and then increases as the cell contact area $A$
contracts. The gradient coefficient $c_{0}$ remains positive, but the main
‘control’ coefficient $r_{0}$ could become negative at a critical value of
$N/A$ (where $g_{2}=k_{B}T/8J$) and cause the cadherin distribution to become
unstable. Note that near this transition point $c_{0}$ takes the value
$c_{0}=2\beta J$.

## Results and comparison with experimental data

In our model, the gas-liquid condensation transition occurs when
$r_{0}(J,N/A)=0$. Given $J\approx 5k_{B}T$ this gives $N/A\approx 0.05$ in a
2D lattice model of cadherin aggregation. This is our first key result, and it
matches well with the in-vitro observation that cadherin clusters form when
their surface density increases past $1100\ \text{cadherins}\
\mu\text{m}^{-2}$ [54] (this is different from the fraction ca.$0.01$ of the
maximum cadherin surface density which they report, because they consider a
much tighter packing of cadherins to within $3$ nm of their neighbours).

Spatial frequency of fluctuations. Near the transition point, the Ginzburg-
Landau action (4) can be approximated by its quadratic terms. The time-
dependence of the concentration fluctuation near the transition point can be
described by the Cahn-Hilliard equation [63]:

$\frac{\partial\phi}{\partial t}=D\nabla^{2}\Big{(}r_{0}\phi-
c_{0}\nabla^{2}\phi\Big{)}\ \ ,$ (6)

where $D$ is the (not yet dimensional) cadherin diffusion coefficient in the
plane. We impose Neumann boundary conditions on a 2D rectangular cell
interface (a reasonable approximation for the lateral surfaces of epithelial
cells for instance), which makes the time- and the spatial coordinates fully
separable, and gives the Cahn-Hilliard solution:

$\displaystyle\phi(x,y,t)=\frac{1}{2}A_{0}(t)+\Sigma_{\mathbf{k}}A_{\mathbf{k}}(0)e^{-D\frac{\mathbf{k}^{2}}{4}\Big{(}r_{0}+\frac{c_{0}\mathbf{k}^{2}}{4}\Big{)}t}\cos{\Big{(}\frac{k_{x}x}{2}\Big{)}}\cos{\Big{(}\frac{k_{y}y}{2}\Big{)}},$
(7)

where $\mathbf{k}=(k_{x},k_{y})$ is the wavevector, directly related to the
numbers $(m_{x},m_{y})$ of peaks along the two spatial directions in our
contact plane: $\mathbf{k}=(2\pi m_{x}/L_{x},2\pi m_{y}/L_{y})$, where the
sizes of the lattice in the x and y directions are $L_{x}$ and $L_{y}$
respectively, with $L_{x}L_{y}=A$.

The size of the fastest growing wavevector, which corresponds to the
oscillation length scale that maximizes the exponential term above, are

$|\mathbf{k}|_{\text{max}}=\sqrt{{-2r_{0}}/{c_{0}}}\ .$ (8)

A range of mode numbers $(m_{x},m_{y})$ satisfy the conditions above.
Assuming, as experiments suggest in Fig. 1, that the punctate adherens
junctions are roughly equidistant from each other in both directions, we find
that the wavenumber of the fastest-growing mode satisfies:

$\displaystyle k_{x,\text{max}}\approx
k_{y,\text{max}}\Rightarrow\frac{m_{x,y}}{L_{x,y}}=\frac{1}{2\pi}\sqrt{-\frac{r_{0}}{c_{0}}}.$
(9)

The total number of clusters within the patch of cell-cell contact area is the
product $M=m_{x}m_{y}$.

Mode destabilisation time. A mode cannot grow until it becomes unstable, that
is, when the second-order terms in the Ginzburg-Landau action become negative
for that value of $k$. Lower $k$ wavevectors become unstable closer to the
density at the transition point ($r_{0}$), while at higher $k$ wavevectors the
increasing interface energy requires a larger negative $r_{0}$ to destabilize
the homogeneous density. Alternatively, large diffuse clusters form before
smaller clusters. Assuming a steady rate change of the contact area, we find
in Supplementary Part F that the largest number of clusters $M$ in the
lattice, for which the combined second order Ginzburg-Landau term is negative,
is proportional to the time $t_{1}$ that has elapsed since the cadherin
density crossed the transition point $r_{0}=0$. Reintroducing dimensional
length scales, we find this linear relation:

$\displaystyle M\approx\Psi(\beta J)\frac{t_{1}|\dot{\Sigma}|}{a^{2}}\ ,$ (10)

where $\dot{\Sigma}$ is the rate of decrease in the cell-cell contact area,
and $\Psi(\beta J)$ is a complicated function obtained from the series
expansion, Eqn. (F.2).

Mode growth time. For the fastest growing mode,
$r_{0}=-c_{0}k_{\text{max}}^{2}/2$, Eqn. (8), so we find that the mode has a
characteristic exponential growth time $t_{2}$ which dictates how fast the
Ginzburg-Landau free energy is minimized. We find this time by substituting
into the exponent in the Cahn-Hilliard solution (7):

$e^{-D\frac{k^{2}}{4}\Big{(}r_{0}+\frac{c_{0}k^{2}}{4}\Big{)}t}\Big{|}_{\text{max}}=e^{D\frac{k^{4}c_{0}}{16}t}$
(11)

Substituting the constants $r_{0}$ and $c_{0}$, and recovering the proper
dimensional length scales via the spacing $a$, we find that the growth time of
the mode becomes:

$t_{2}=\alpha\frac{A^{2}a^{2}}{2^{6}\pi^{4}M^{2}Dg_{2}\beta^{2}J^{2}}\ ,$ (12)

i.e. inversely proportional to $M^{2}$, with $\alpha$ a proportionality
constant of order $1$.

Density of punctate adherens junctions. The total time for a mode to first
become unstable and then reach the minimum in the Ginzburg-Landau free energy
landscape is of the order:

$t_{\text{tot}}=t_{1}+t_{2}=K_{1}M+K_{2}M^{-2}$ (13)

where $K_{1}$ and $K_{2}$ are the proportionality constants in Eqs. (10,12).
The total time is minimized when the number of clusters across the contact
area is:

$\displaystyle\frac{dt_{\text{tot}}}{dM}$
$\displaystyle=K_{1}-2K_{2}M^{-3}=0\Rightarrow
M^{*}=\Big{(}\frac{2K_{2}}{K_{1}}\Big{)}^{1/3}.$ (14)

We make the proportionality constants explicit and find that the number of
adherens junctions per unit area of cell-cell interface is given by

$n_{\text{s}}=\frac{M^{*}}{a^{2}A}=\Big{(}\alpha\frac{\Psi(\beta
J)}{2^{5}\pi^{4}a^{4}Dg_{2}\beta^{2}J^{2}}\frac{|\dot{\Sigma}|}{\Sigma}\Big{)}^{1/3}$
(15)

This is the second main prediction of this paper.

Parameter | Name | Value | Uncertainty | References
---|---|---|---|---
Cadherin lattice |  |  |  |
spacing | a | 7 nm |  $\pm$ 1 nm | [64, 50, 51, 54]
Cadherin diffusion |  |  |  |
coefficient (macroscopic) | D |  $2.6\cdot 10^{-3}\mu\text{m}^{2}/$s |  $\pm 1.1\cdot 10^{-3}\mu\text{m}^{2}/$s | [65]
Cadherin diffusion |  |  |  |
coefficient (microscopic) | D |  $3.3\cdot 10^{-3}\mu\text{m}^{2}/$s |  $\pm 2.9\cdot 10^{-3}\mu\text{m}^{2}/$s | [65]
Cis interaction energy | J | $5k_{B}T$ | $\pm 2k_{B}T$ | [50, 61]
Fractional shrinkage of |  |  |  |
cell-cell contact area | $|\dot{\Sigma}|/\Sigma$ | $\approx 20\%\text{min}^{-1}$ | $\pm 5\%\text{min}^{-1}$ | Figure 2 [66]
Density at the phase transition | N/A | 0.05 | $\pm 0.01$ | (see above and [54])
Arbitrary exponential growth parameter | $\alpha$ | 1-10 | $*$ | NB$\dagger$
Table 1: Summary of the physiological values used in the model. Note the large
standard deviation in the cadherin diffusion constants. This arises due to the
different modes of cadherin movement [65]; however, the uncertainty on the
average value is much less than this standard deviation would suggest.
$\dagger$ NB: Depends on the tolerance on how far the coarsening has
progressed. Here we choose a factor of $e$ (ca. $63\%$ growth), whereas
simulations with a $1\%$ tolerance [67] give a value of $\alpha$ closer to 10.
It seems clear that cadherin clustering should become visible before
coarsening completes.

To make an experimental comparison, we use the aggregated cadherin spacing $a$
= 6 nm [64, 50, 51] (slightly smaller than the crystal lattice size seen by
[51] to account for the possible tighter packing observed by [54]), diffusion
constant $D=3\cdot 10^{-3}\mu\text{m}^{2}/$s [65], density at transition of
$N/A=0.04$ (see above), $J=5k_{B}T$, and proportionality constant $\alpha=1$
for the sake of simplicity (see Table 1). The fractional shrinking rate of a
patch of the cell-cell contact area $(|\dot{\Sigma}|/\Sigma)$ depends on the
cellular process and the type of cell.

Figure 2: Evolution of cadherin density on the apical edge of a dividing
epithelial cell during cytokinesis, reproduced with permission from [66].
Cadherins are labelled with E-cad::GFP fluorescent markers; the second panel
(kymograph) shows the density along the [a-b] cross-section, illustrating how
the interface area contracts and gets populated by increasingly inhomogeneous
cadherin clusters. Aggregation begins only after $200$s; during this time, the
length of the apical edge of the new contact area between daughter cells
reduces at a rate of $|\dot{\Sigma}|/\Sigma\approx 20\%\ \text{min}^{-1}$.

One classic example which requires cadherin clusters to form occurs during
cytokinesis at the end of cell division [68]. Kymographs of the cross-section
of the a dividing epithelial cell [66, 69], see Fig. 2, show that the
fractional change in the area of the cell interface is
$|\dot{\Sigma}|/\Sigma\approx 20\%$ per minute. For the case of epithelial
cell division, therefore, we find $n_{\text{s}}\approx 28\pm 4\
\text{clusters}\ \mu\text{m}^{-2}$, assuming a $40\%$ uncertainty for the
cadherin diffusion coefficients (macroscopic uncertainty value in Table 1)..
This appears to match well with the number density of punctate adhesions
observed by [24] in epithelial tissues seen in Fig. 1. Both experimental and
computational studies have also reported the characteristic cluster size
values of $33$ cadherins [19], which gives the estimate of average cadherin of
the correct order of ca.$1000\ \text{cadherins}\ \mu\text{m}^{-2}$ [54]. Even
though there might be some uncertainty in the values of physiological
constants used to evaluate Eqn. (15), the cube-root dependence of the number
density on these constants makes a large error unlikely: any one parameter
value would need to be off by a factor of $1000$ for there to be $10$ times
fewer or more punctate adhesions. Note that we can also use this figure to
estimate the total growth and destabilisation time to be of the order of 5
seconds, once the cadherin distribution becomes unstable. This would account
for the quite sharp transition to a higher cadherin instensity in the
kymograph at the time indicated by the arrow.

## Conclusions

In this paper, we built a 2D model for the aggregation of adhesion units with
a short-distance attractive interaction, which is applicable to all cell
contact areas if the problem is reduced down to a sufficiently small and
uniform patch (locally, with zero curvature and applied force variation), so
we expect our results to be more universally valid. The advantage of treating
cadherin adhesion complexes as a specific example is that there is a large
body of experimental and computational work from which to test our results.
Our predictions for the transition density of cadherins above which clusters
could form, as well as for the number density of cadherin clusters, were
independent of each other. Together, they strongly suggest that cadherin
density-depedent cluster nucleation initially occurs via a gas-liquid phase
transition.

Previous studies have analytically explained how cadherin trans bonds can help
set up large scale adherens junctions[23], and new work has computationally
shown that cadherin cis-bonds help develop punctate adherens junctions[61]. In
this work. we have laid out for the first time an analytical explanation for
how punctate cadherin adherens junctions can form.

While we looked at the case of cadherin nucleation, more generally, any
distribution of attractively-coupled cell membrane molecules in the low
concentration limit can undergo a gas-liquid (condensation) phase transition
in the form of density-dependent aggregation. Because of this, we could apply
this method to the more general problem of the formation of membrane-bound
organelles.

### Data access

All data generated or analysed during this study are included in this
published article [and its supplementary information files].

### Author contribution

Both authors conceived the idea, carried out different elements of data
analysis and wrote the paper.

### Acknowledgements

We are grateful to R. Zaidel-Bar as well as to T. Lecuit for giving access to
their experimental results. This work has been funded by BBRSC DTP Cambridge
(grant no. EP/M508007/1).

## References

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