# Tunable shape oscillations of adaptive droplets

Tim Dullweber  European Molecular Biology Laboratory, Meyerhofstraße 1, 69117
Heidelberg
Department of Physics and Astronomy, Heidelberg University, 69120 Heidelberg
Roman Belousov  European Molecular Biology Laboratory, Meyerhofstraße 1, 69117
Heidelberg
Anna Erzberger<EMAIL_ADDRESS>European Molecular Biology Laboratory,
Meyerhofstraße 1, 69117 Heidelberg
Department of Physics and Astronomy, Heidelberg University, 69120 Heidelberg

###### Abstract

Living materials adapt their shape to signals from the environment, yet the
impact of shape changes on signal processing and the associated feedback
dynamics remain unclear. We derive coarse-grained equations for droplets that
adjust their interfacial tension in response to signals exchanged at contact
surfaces, from the microscopic biophysics of adhesion and signaling. We find
that droplet pairs exhibit symmetry-breaking, excitability, and oscillations.
The underlying critical points reveal novel mechanisms for physical signal
processing through shape adaptation in soft active materials.

††preprint: APS/123-QED

Dynamic and non-trivial geometries are hallmarks of soft active matter,
because intrinsic stress fields drive autonomous shape changes in deformable
materials [1, 2, 3, 4, 5]. In turn, boundary geometry can influence stresses
and material properties, or determine how macroscopic work is extracted from
microscopic sources of activity [6, 7]. Some complex materials process signals
and adaptively respond to their environment [8, 9, 10]. Geometry-dependent
feedback arises when signal processing depends on the system’s shape.

In particular living materials possess internal degrees of freedom that adjust
their mechanical properties in response to peripheral signals. In cells,
signals trigger biochemical processes including the regulation of gene
expression that control the molecular composition in the bulk and at the
surface [11]. Inhibitory signaling interactions between neighboring cells for
example lead to the spontaneous symmetry-breaking of such internal states,
giving rise to distinctly shaped cell types [12, 13]. The resulting
mechanochemical feedback dynamics govern the spatial organisation of diverse
multicellular systems [14, 15, 16].

We propose that geometry-dependent feedback effects generically underlie the
capacity of soft active systems to autonomously solve tasks including
locomotion [17, 18, 19, 20], self-healing [21, 22], and the self-organisation
of complex structures [23, 24, 25]. Yet, how internal cellular states interact
with shape dynamics is an open question [26, 27, 25], and more generally, how
the phase space of adaptive shape-changing materials depends on geometry is
unclear.

Uncovering the theoretical principles governing the rich physics of adaptive
active matter systems requires minimal, tractable paradigms. Here, we consider
adaptive droplets that change their surface tension in response to signals
received at contact surfaces with other droplets or substrates. Particularly
interesting dynamics appear for mutually inhibitory interactions, i.e. when
signals received by a droplet reduce its own capacity to send signals. To
analyse how nonlinear signal processing interplays with fundamental
nonlinearities in geometrical relations, we derive a minimal set of equations
governing the macroscopic droplet states, controlled by two dimensionless
feedback parameters. These equations are consistent with microscopic reaction-
diffusion dynamics of signaling and adhesion molecules (SM [28], Sec. .1). We
show that coupling between active mechanics and signaling creates a variety of
nonlinear phenomena, including bistability, excitability, and diverse
oscillations of droplet shapes and internal states, which arise from a saddle-
node pitchfork codimension-2 bifurcation point.

##### Adaptive Young-Laplace droplets.—

We consider a pair of Young-Laplace droplets with interfacial areas governed
by the conjugate uniform surface tensions at fixed volumes [Fig. 1(a)]. The
total surface energy for a pair of identical droplets is

$E=\gamma_{\rm{c}}A_{\rm{c}}+2\gamma_{\rm{f}}A_{\rm{f}},$ (1)

in which $\gamma_{\rm{c}}$ and $\gamma_{\rm{f}}$ are the surface tensions of
the contact interface $A_{\rm{c}}$ and the outer surface area $A_{\rm{f}}$,
respectively. For positive interfacial tensions, the surface energy is minimal
when both droplets acquire a spherical-cap shape with contact area

$\displaystyle\dfrac{A_{\rm{c}}}{A_{0}}=\left[1-\left(\frac{\gamma_{\rm{c}}}{2\gamma_{\rm{f}}}\right)^{2}\right]\left[\dfrac{2}{\left(2-\dfrac{\gamma_{\rm{c}}}{2\gamma_{\rm{f}}}\right)\left(1+\dfrac{\gamma_{\rm{c}}}{2\gamma_{\rm{f}}}\right)^{2}}\right]^{\frac{2}{3}}$
(2)

relative to the reference area $A_{0}=(3V/2)^{2/3}\pi^{1/3}$ determined by the
conserved volumes $V_{1}=V_{2}=V$ [Fig. 1(c)].

Figure 1: (a) We consider equilibrium shapes of Young-Laplace droplet pairs,
which adapt the tension $\gamma_{\rm{c}}$ at their contact surface
$A_{\rm{c}}$ to their internal states $u_{1},u_{2}$. The droplets have
conserved volumes $V_{1},V_{2}$ and uniform outer surface tensions
$\gamma_{\rm{f},1},\gamma_{\textrm{f},2}$ at the outer surfaces
$A_{\textrm{f},1},A_{\textrm{f},2}$. (b) The signal susceptibility $\chi$ and
the adaptive adhesion coefficient $\gamma_{\rm{A}}$ govern the feedback
between shape and signal processing (c) The equilibrium contact area depends
on tension ratio $\gamma_{\rm{c}}/2\gamma_{\rm{f}}$—shown here for equal
volumes and outer surface tensions [Eq. (2)]—and sets the magnitude of the
exchanged signals [Eq. (5)]. (d) The internal states evolve according to
mutually inhibitory interactions [Eq. (4)-(5)], with a sigmoidal response
function $\sigma(s_{ij})=s_{ij}^{4}/(s_{ij}^{4}+1)$.

We take the adhesion between the droplets to depend on the signals exchanged
at their shared surface. Therefore, to each droplet $i\in\\{1,2\\}$ we assign
a dimensionless _internal state_ $u_{i}\in[0,1]$, a variable that increases in
response to received signals in a saturating manner [Fig. 1(d),[29, 16, 30]].
When the internal state drives active processes that increase the adhesion
between the droplets, a first-order relation linking the tension at the
droplet-droplet interface and the internal states can be written as

$\displaystyle\gamma_{\rm{c}}=\gamma_{0}-\gamma_{\rm{A}}u_{1}u_{2},$ (3)

in which the second term contains the active contributions in response to
signaling, while $\gamma_{0}$ contains all other components of the interfacial
tension. A variety of active microscopic processes can modulate the effective
tension at the surface of cells, including the generation of stresses by
molecular motors within polymerizing cytoskeletal networks [31, 32, 33] and
biochemical regulation of adhesion [34, 35, 36]. We derive Eq. (3) considering
a signal-driven production of adhesive molecules within the droplets’ bulk and
the steady-state reaction kinetics of adhesion complexes forming at the
contact interface ([28], Sec. .2). The product of internal states in the
adaptive tension arises for mass-action reaction kinetics, and a linear
dependence of adhesion molecule production in the bulk on the internal states.
The _adaptive adhesion coefficient_ $\gamma_{\rm{A}}$ captures the strength of
the coupling between the internal states and the interfacial tension. Because
adhesion reduces the tension, the adaptive term is always negative. It can be
expressed as $\gamma_{\rm{A}}=\epsilon/\lambda^{2}$ in terms of an energy per
adhesion complex $\epsilon$ and a length scale $\lambda$ dependent on the rate
of turnover at the bulk-surface interface (Eq. (76) in [28]). Similar
relationships between the molecular composition—for instance the concentration
of molecular motors—and stresses at the cell surface have been used
successfully to describe tension fields and predict associated flows and shape
changes in living systems [37, 38, 39]. The adaptive tension vanishes in the
absence of signaling interactions and it always acts to increase the
interfacial area $A_{\rm{c}}(\gamma_{0})\leq A_{\rm{c}}(\gamma_{\rm{c}})$ [Eq.
(2)] up to the limit set by the saturation of the internal states. A very
large adaptive adhesion coefficient leads to a buckling instability of the
interface when $\gamma_{\rm{c}}<0$ – an increase in the interface area then
decreases the surface energy [40, 5]. Our analysis focuses on the regime
$\gamma_{\rm{A}}<\gamma_{0}$ in which Eq. (2) is valid.

The states of the droplets $u_{i}$ are taken to evolve according to a generic
mutually inhibitory signaling interaction [16, 30]

$\displaystyle\tau_{\rm{u}}\frac{du_{i}}{dt}=\sigma(s_{ij})-u_{i},$ (4)

in which $\sigma(s_{ij})$ is an increasing sigmoidal response function to a
signal $s_{ij}$ received by droplet $i$ from droplet $j$ [Fig. 1(d)]. We use a
Hill function $\sigma(s_{ij})=s_{ij}^{h}/(s_{ij}^{h}+1)$ with Hill coefficient
$h=4$ [41, 42]. The magnitude of $s_{ij}$ depends on the internal state of the
sending droplet $u_{j}$ and—for mutually inhibitory coupling—must decrease
with $u_{j}$. We use in the following a first-order relation taking into
account a linear dependence of the exchanged signals on the contact area [43,
44]

$s_{ij}=\chi\frac{A_{\rm{c}}}{A_{0}}(1-u_{j})$ (5)

with a dimensionless _signal susceptibility_ parameter $\chi$. Equations
(4)–(5) can be derived from reaction-diffusion dynamics of biochemical
signaling molecules ([28], Sec. .3.2) in which $u_{i}$ is defined as a
rescaled and normalized concentration of a regulator molecule, such as a
transcription factor. These microscopic equations relate $\chi$ to the kinetic
rates associated with the turnover, binding, and processing of signaling
molecules. [Eq. (43),(74) in [28]]. The linear dependence of transmitted
molecular signals on the contact area is valid when diffusion across the
contact line and the number of molecules lost through biochemical processes at
the surface are negligible.

Provided that the adaptive droplet pairs relax to their equilibrium
configuration fast compared to the signaling time scale $\tau_{\rm{u}}$ and
the corresponding change in the interfacial tension, the coupled dynamics of
the doublet are governed by Eqs. (2)–(5). Two coupling coefficients
characterize the feedback between the system’s geometry and the internal
states: the adaptive adhesion coefficient $\gamma_{\rm{A}}$ controls the
effect of signaling on the surface mechanics and thereby shape, and the signal
susceptibility $\chi$ captures how transmitted signals depend on the contact
area [Fig. 1(b)].

Figure 2: Nonlinear dynamics of adaptive Young-Laplace droplet pairs. (a) The
droplet pair state diagram shows that tension adaptation increases the
parameter regime (green) associated with spontaneous symmetry-breaking due to
mutual inhibition (PF-: supercritical pitchfork PF+: subcritical pitchfork),
and leads to self-sustained oscillations of signals and droplet shapes (color
gradient and contour lines denote the oscillation period $T$). The oscillatory
regime is surrounded by saddle-heteroclinic (SHET) and Hopf (H) bifurcation
lines that originate from a saddle-node pitchfork codimension-2 point (SP).
Bottom panel: Enlarged view of the SP point environment shows saddle-node (SN)
and cusp bifurcations that preserve stable attractor structures. The reference
susceptibility is the critical value in the absence of adaptive tension
($\chi_{0}=\left.\chi_{\rm{PF}}\right|_{\gamma_{\rm{A}}=0}$). (b) The state
diagram of a single adherent droplet on a signal-transmitting substrate
contains a bistable regime bounded by two SN lines that originate from a
codimension-2 cusp point. Inset: steady-state internal state as a function of
the signal susceptibility (along gray dashed line) (c) Phase portraits for
parameter values marked with gray symbols (filled black circle: stable steady
state, filled gray circle: saddle, open circle: unstable steady state, rose
line: trajectory in the excitable regime, red line: heteroclinic, black line:
limit cycle). (d-e) Oscillation amplitudes decrease with waveforms changing
from relaxation-like (near the SHET line) to sinusoidal (near the Hopf line)
for increasing $\chi$. Parameter values given in [28], Tab. 2.

##### Adaptive tension promotes symmetry-breaking.—

Strong mutually inhibitory interactions generically lead to spontaneous
symmetry-breaking, whereby initially small differences between interacting
units diverge to high- and low-value steady states [12]. Linear stability
analysis, numerical continuation, and simulations of Eqs. (2)–(5) reveal that
pairs of adaptive droplets undergo symmetry-breaking of internal states via a
line of supercritical pitchfork bifurcations (PF) in the state diagram spanned
by the signal susceptibility $\chi$ and the adaptive adhesion coefficient
$\gamma_{\rm{A}}$ [Fig. 2(a-b)]. Below the critical value $\chi_{\rm{PF}}$,
inhibition is not sufficient to produce symmetry-breaking, and the two
droplets converge to identical internal states near zero in a configuration
with a small contact. The critical susceptibility scales approximately
inversely with the interfacial area $\chi_{\rm{PF}}\sim A_{0}/A_{c}$ ([28],
Sec. .3.2). Thus, adhesive tension adaptation promotes symmetry-breaking: the
active term in Eq. (3) transiently expands the contact area, thereby lowering
the threshold susceptibility. This shows that adaptive shape changes enable
state transitions beyond the parameter regime attainable in static
configurations.

##### Tunable self-sustained oscillations.—

At large adaptive tension, the coupling between signaling and interface
geometry leads to diverse self-sustained oscillations in the shape of the
droplet pair and the internal states. Specifically, stable symmetric and
symmetry-broken states are separated by an oscillatory regime, bounded by Hopf
(H) and saddle heteroclinic (SHET) bifurcation lines [Fig. 2(a-b)]. These
lines originate from a saddle-node pitchfork bifurcation point (SP)—a
codimension-2 bifurcation at which the PF line tangentially intersects with a
saddle-node (SN) bifurcation line [[28], Fig. 13] [45]. In this region, the
coupling between interface area and transmitted signals [Eqs. (3)–(5)] gives
rise to a bistability between small and large contact area configurations,
which competes with the tendency of the droplet pair to undergo symmetry-
breaking of internal states. A negative feedback arises between the product of
states $u_{1}u_{2}$—which increases the contact area via Eq. 3—and the contact
area across which inhibition occurs.

This area bistability is exhibited also by single adherent droplets on signal-
transmitting substrates [[28], Sec. .3.1], where the state diagram contains
two SN lines that originate from a codimension-2 cusp point [Fig. 2(c) and
[28], Sec. .3.1]. Above a critical susceptibility, the positive feedback
between signaling and adhesion enables two stable configurations: since the
contact area limits the amplitude of adhesion-inducing signals from the
substrate, a droplet with a small contact area remains weakly adhesive,
whereas a large contact area permits the transmission of strong signals,
promoting substrate wetting. At large susceptibilities, the bistability is
lost, and the configuration with large contact area remains the only steady
state.

In the two-dimensional phase space of the droplet pair, the SN bifurcation
associated with the area bistability produces a saddle point and an unstable
fixed point instead of the pair of stable and unstable fixed points observed
in the single-droplet system [Fig. 2(b) quatrefoil]. When
$\gamma_{\rm{A}}>\gamma_{\rm{A}}^{\rm{SP}}$ and $\chi$ reaches the critical
susceptibility $\chi_{\rm{PF}}$, the inhibitory signals induce symmetry-
breaking and the unstable fixed point undergoes a subcritical pitchfork
bifurcation, producing a saddle and two new unstable fixed points [Fig. 2(b)
star]. In this regime the droplet pair is excitable: fluctuations moving the
internal states beyond the separatrices, which connect the saddle to the
unstable fixed points, trigger a large increase of both internal states and
the contact area $A_{\rm{c}}$, followed by transient symmetry-breaking [Fig.
2(b) star and Movie 1]. Increasing $\chi$ shortens the distance between the
uniform stable fixed point and the saddle, thus lowering the excitation
threshold until the two points collide at the SHET line and give rise to a
pair of heteroclinic orbits that connect the resulting transversely stable,
nonhyperbolic point to the second saddle point [Fig. 2(b) plus]. This
nonhyperbolic point is destroyed as the heteroclinic orbits bifurcate into two
symmetric stable limit cycles [Fig. 2(b) pentagon], which remain the only
stable attractors of the system. Thus, cycles appear once transmitted signals
are strong enough to induce symmetry-breaking, which in turn lowers the
adhesion—and thereby the contact area—sufficiently to reduce signals below the
symmetry-breaking threshold. In turn, the product of states [Eq. (75)]
increases again, thereby driving adhesion, contact area, and signal amplitude
back above the threshold. These results illustrate how the droplets’ geometry
can act as a form of memory—here encoded in configurations of varying contact
size—and enable the generation of complex temporal dynamics including
excitability and self-sustained oscillations.

Depending on the two feedback parameters, droplet oscillations exhibit a range
of temporal profiles. Near the SHET line the droplet pair exhibits relaxation-
type oscillations in which it spends a large fraction of the cycle in small-
area configurations with nearly identical states, interrupted by spikes in the
contact area $A_{\rm{c}}$ and rapid, transient symmetry-breaking [Fig. 2(d)].
The oscillation period diverges as $\chi$ approaches $\chi_{\rm{SHET}}$ due to
the ghost of the destroyed saddle point that critically slows down the limit-
cycle phase when passing through its vicinity [Fig. 2(a)]. With increasing
$\chi$, the time-averaged difference between the internal states increases and
the oscillation amplitudes decrease, reaching near-sinusoidal waveforms in
states and contact area close to the Hopf bifurcation line, where the limit
cycles smoothly contract into symmetry-broken fixed points [Fig. 2(b,d-e)].

Figure 3: Nonlinear dynamics of asymmetric droplet pairs. (a) Differences in
droplet volumes $\delta V/\bar{V}=\\{0.25,0.5\\}$ (blue) or outer tensions
$\delta\gamma_{\rm{f}}/\bar{\gamma_{\rm{f}}}=\\{0.25,0.5\\}$ (brown) shift the
SP point and associated bifurcation lines in the state diagram. (b) A
difference in signal susceptibilities $\delta\chi/\bar{\chi}=0.05$ (i.e.
$\chi_{1}/\chi_{2}\approx 1.1$) splits the SP point into a pair of Bogdanov-
Takens bifurcation points (BT), a non-central homoclinic to saddle-node
bifurcation (NCH) and associated bifurcation lines reconnect in a non-trivial
fashion. Inlet shows the state diagram close to the second BT point
($\gamma_{\rm{A}}/\gamma_{0}\in[0.3896,0.403]$,
$\chi/\chi_{0}\in[0.5388,0.5441]$). Note that NCH and BT are connected by a
homoclinic (Hom). (HSN: Saddle-node homoclinic). (c) Phase portraits for
parameter values marked with gray symbols in (b). (filled black circle: stable
steady state, filled gray circle: saddle, open circle: unstable steady state,
rose line: trajectory in the excitable regime, thick black line: limit cycle).
Parameter values given in [28], Tab. 2.

##### Asymmetric droplets.—

The state-diagram structure associated with the SP point arises for identical
droplets. While such state-space structures have been found and experimentally
characterized for instance in optical cavities [45], most physical systems
exhibit non-negligible variations in their properties. Differences in the
properties of the interacting droplets change the state diagram shown in Fig.
2(a). For unequal droplet volumes $V_{1,2}=\bar{V}\pm\delta V$, symmetry-
breaking and oscillatory dynamics emerge at a larger signaling susceptibility
$\chi$ than in pairs of identical droplets, whereas a difference in the outer
surface tensions
$\gamma_{\rm{f},1,2}=\bar{\gamma_{\rm{f}}}\pm\delta\gamma_{\rm{f}}$ promotes
symmetry-breaking and oscillations at lower susceptibilities due to partial
internalization resulting in larger equilibrium contact areas [Fig. 3(a)].

Tension and volume asymmetry do not favour any droplet to reach a higher or
lower internal state, because the signaling properties of each droplet remain
unaffected, and thus the topology of the state space is preserved. In
contrast, a difference in the signaling susceptibility
$\chi_{1,2}=\bar{\chi}\pm\delta\chi$ splits the SP point into two Bogdanov-
Takens (BT) codimension-2 points, and the SHET line into two homoclinics (Hom)
and a saddle-node homoclinic (HSN) bifurcation line emerging from a non-
central homoclinic to saddle-node (NCH) [Fig. 3(b)]. Accordingly, the limit
cycle and the corresponding symmetry-broken state, in which the less
susceptible droplet maintains the lower internal state, require lower values
of $\chi$ and $\gamma_{\rm{A}}$ than the inverse symmetry-broken states. This
allows for parameter regimes with single limit cycles [Fig. 3(c) hexagon] or
coexistence with stable fixed points [Fig. 3(c) 4-pointed star]—contrary to
the case of identical susceptibilities [Fig. 3(c) square and star].
Heterogeneous material properties can thus produce an even wider spectrum of
dynamics.

##### Conclusions

Considering adaptive, interacting droplets, we obtained a tractable set of
equations that reveal the rich physics arising in signal-processing active
materials. In particular, we find that coupling between the geometry of
droplet interfaces and the signals exchanged at these physical contacts enable
robust symmetry-breaking, excitability, and self-sustained oscillations that
can be tuned from sinusoidal to relaxation-like waveforms.

The critical points and associated phase-space structures we identify thus
reveal how fundamental geometrical relations permit shape-adapting systems to
produce a wide range of time-encoded outputs with few degrees of freedom.
Indeed, dynamical features of signaling levels can program distinct cell
fates[46, 47, 48], suggesting a role for shape-dependent feedback in the self-
organisation of multicellular structures [49, 50].

Our findings will enable the discovery of new collective phenomena in active
signal-processing materials, where mechanical feedback can drive spontaneous
patterning and wave dynamics[51, 52, 53]. In particular, investigating
fluctuation-induced effects in the excitable regimes will reveal when the
large area deviations we report trigger topological transitions governing
global rheological properties of materials such as tissues [54, 55]. Analysing
these collective dynamics will uncover how geometry-dependent feedback
produces novel modes of self-organisation in adaptive materials.

##### Acknowledgements

We gratefully acknowledge insightful discussions and valuable feedback from
Florian Berger, Erwin Frey, Isabella Graf, Jeremy Gunawardena, Adrian Jacobo,
Thomas Quail, Ulrich Schwarz, Alejandro Torres-Sánchez, Falko Ziebert and all
members of the Erzberger Group. We acknowledge funding by the EMBL and TD was
supported by a Joachim-Herz Add-on fellowship for interdisciplinary life
science.

## References

  * Mietke  _et al._ [2019] A. Mietke, F. Jülicher, and I. F. Sbalzarini, Proceedings of the National Academy of Sciences 116, 29 (2019).
  * Salbreux and Jülicher [2017] G. Salbreux and F. Jülicher, Physical Review E 96, 032404 (2017).
  * Marchetti  _et al._ [2013] M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Reviews of modern physics 85, 1143 (2013).
  * Shankar  _et al._ [2022] S. Shankar, A. Souslov, M. J. Bowick, M. C. Marchetti, and V. Vitelli, Nature Reviews Physics 4, 380 (2022).
  * Binysh  _et al._ [2022] J. Binysh, T. R. Wilks, and A. Souslov, Science advances 8, eabk3079 (2022).
  * Ray  _et al._ [2023] S. Ray, J. Zhang, and Z. Dogic, Physical Review Letters 130, 238301 (2023).
  * Araújo  _et al._ [2023] N. A. Araújo, L. M. Janssen, T. Barois, G. Boffetta, I. Cohen, A. Corbetta, O. Dauchot, M. Dijkstra, W. M. Durham, A. Dussutour, _et al._ , Soft matter 19, 1695 (2023).
  * McEvoy and Correll [2015] M. A. McEvoy and N. Correll, Science 347, 1261689 (2015).
  * Zhao  _et al._ [2023] H. Zhao, A. Košmrlj, and S. S. Datta, arXiv preprint arXiv:2301.12345 (2023).
  * Ziepke  _et al._ [2022] A. Ziepke, I. Maryshev, I. S. Aranson, and E. Frey, Nature communications 13, 6727 (2022).
  * Alberts  _et al._ [2022a] B. Alberts, R. Heald, A. Johnson, D. Morgan, M. Raff, K. Roberts, and P. Walter,  _Molecular Biology of the Cell_ (WW Norton & Company, 2022) Chap. 15.
  * Pisarchik and Hramov [2022] A. N. Pisarchik and A. E. Hramov,  _Multistability in Physical and Living Systems_ , Vol. 2 (Springer, 2022) Chap. 3.
  * Sjöqvist and Andersson [2019] M. Sjöqvist and E. R. Andersson, Developmental biology 447, 58 (2019).
  * Dullweber and Erzberger [2023] T. Dullweber and A. Erzberger, Current Opinion in Systems Biology , 100445 (2023).
  * Sprinzak and Blacklow [2021] D. Sprinzak and S. C. Blacklow, Annual review of biophysics 50, 157 (2021).
  * Erzberger  _et al._ [2020] A. Erzberger, A. Jacobo, A. Dasgupta, and A. Hudspeth, Nature physics 16, 949 (2020).
  * Luo  _et al._ [2023] D. Luo, A. Maheshwari, A. Danielescu, J. Li, Y. Yang, Y. Tao, L. Sun, D. K. Patel, G. Wang, S. Yang, _et al._ , Nature 614, 463 (2023).
  * Shah  _et al._ [2021] D. S. Shah, J. P. Powers, L. G. Tilton, S. Kriegman, J. Bongard, and R. Kramer-Bottiglio, Nature Machine Intelligence 3, 51 (2021).
  * Venturini  _et al._ [2020] V. Venturini, F. Pezzano, F. Catala Castro, H.-M. Häkkinen, S. Jiménez-Delgado, M. Colomer-Rosell, M. Marro, Q. Tolosa-Ramon, S. Paz-López, M. A. Valverde, _et al._ , Science 370, eaba2644 (2020).
  * Bodor  _et al._ [2020] D. L. Bodor, W. Pönisch, R. G. Endres, and E. K. Paluch, Developmental cell 52, 550 (2020).
  * Terryn  _et al._ [2021] S. Terryn, J. Langenbach, E. Roels, J. Brancart, C. Bakkali-Hassani, Q.-A. Poutrel, A. Georgopoulou, T. G. Thuruthel, A. Safaei, P. Ferrentino, _et al._ , Materials Today 47, 187 (2021).
  * Ajeti  _et al._ [2019] V. Ajeti, A. P. Tabatabai, A. J. Fleszar, M. F. Staddon, D. S. Seara, C. Suarez, M. S. Yousafzai, D. Bi, D. R. Kovar, S. Banerjee, _et al._ , Nature physics 15, 696 (2019).
  * Sun  _et al._ [2023] G. Sun, R. Zhou, Z. Ma, Y. Li, R. Groß, Z. Chen, and S. Zhao, Nature Communications 14, 3476 (2023).
  * Gov [2018] N. Gov, Philosophical Transactions of the Royal Society B: Biological Sciences 373, 20170115 (2018).
  * Paluch and Heisenberg [2009] E. Paluch and C.-P. Heisenberg, Current Biology 19, R790 (2009).
  * Linding and Klipp [2021] R. Linding and E. Klipp, Current Opinion in Systems Biology 27, 100354 (2021).
  * Prasad and Alizadeh [2019] A. Prasad and E. Alizadeh, Trends in biotechnology 37, 347 (2019).
  * Dullweber  _et al._ [2023] T. Dullweber, R. Belousov, and A. Erzberger, Supplemental Material contains derivation of Eq.(2)-(5) from microscopic reaction-diffusion equations, a summary of the biological background and details of linear stability analysis, continuation and other numerical methods. (2023).
  * Herszterg  _et al._ [2023] S. Herszterg, M. de Gennes, S. Cicolini, A. Huang, C. Alexandre, M. Smith, H. Araujo, J.-P. Vincent, and G. Salbreux, bioRxiv , 2023 (2023).
  * Corson  _et al._ [2017] F. Corson, L. Couturier, H. Rouault, K. Mazouni, and F. Schweisguth, Science 356, eaai7407 (2017).
  * Sitarska and Diz-Muñoz [2020] E. Sitarska and A. Diz-Muñoz, Current opinion in cell biology 66, 11 (2020).
  * Chugh  _et al._ [2017] P. Chugh, A. G. Clark, M. B. Smith, D. A. Cassani, K. Dierkes, A. Ragab, P. P. Roux, G. Charras, G. Salbreux, and E. K. Paluch, Nature cell biology 19, 689 (2017).
  * Nambiar  _et al._ [2009] R. Nambiar, R. E. McConnell, and M. J. Tyska, Proceedings of the National Academy of Sciences 106, 11972 (2009).
  * Alberts  _et al._ [2022b] B. Alberts, R. Heald, A. Johnson, D. Morgan, M. Raff, K. Roberts, and P. Walter,  _Molecular Biology of the Cell_ (WW Norton & Company, 2022) Chap. 19.
  * Schwarz and Safran [2013] U. S. Schwarz and S. A. Safran, Reviews of Modern Physics 85, 1327 (2013).
  * Maître  _et al._ [2012] J.-L. Maître, H. Berthoumieux, S. F. G. Krens, G. Salbreux, F. Jülicher, E. Paluch, and C.-P. Heisenberg, science 338, 253 (2012).
  * Bergert  _et al._ [2015] M. Bergert, A. Erzberger, R. A. Desai, I. M. Aspalter, A. C. Oates, G. Charras, G. Salbreux, and E. K. Paluch, Nature cell biology 17, 524 (2015).
  * Tjhung  _et al._ [2012] E. Tjhung, D. Marenduzzo, and M. E. Cates, Proceedings of the National Academy of Sciences 109, 12381 (2012).
  * Mayer  _et al._ [2010] M. Mayer, M. Depken, J. S. Bois, F. Jülicher, and S. W. Grill, Nature 467, 617 (2010).
  * Bormashenko [2017] E. Y. Bormashenko,  _Physics of wetting: phenomena and applications of fluids on surfaces_ (Walter de Gruyter GmbH & Co KG, 2017).
  * Collier  _et al._ [1996] J. R. Collier, N. A. Monk, P. K. Maini, and J. H. Lewis, Journal of Theoretical Biology 183, 429 (1996).
  * Binshtok and Sprinzak [2018] U. Binshtok and D. Sprinzak, Molecular Mechanisms of Notch Signaling , 79 (2018).
  * Khait  _et al._ [2016] I. Khait, Y. Orsher, O. Golan, U. Binshtok, N. Gordon-Bar, L. Amir-Zilberstein, and D. Sprinzak, Cell reports 14, 225 (2016).
  * Shaya  _et al._ [2017] O. Shaya, U. Binshtok, M. Hersch, D. Rivkin, S. Weinreb, L. Amir-Zilberstein, B. Khamaisi, O. Oppenheim, R. A. Desai, R. J. Goodyear, _et al._ , Developmental cell 40, 505 (2017).
  * Blackbeard  _et al._ [2012] N. Blackbeard, S. Osborne, S. O’Brien, and A. Amann, arXiv preprint arXiv:1210.4484 (2012).
  * Nandagopal  _et al._ [2018] N. Nandagopal, L. A. Santat, L. LeBon, D. Sprinzak, M. E. Bronner, and M. B. Elowitz, Cell 172, 869 (2018).
  * Viswanathan  _et al._ [2021] R. Viswanathan, J. Hartmann, C. Pallares Cartes, and S. De Renzis, The EMBO Journal 40, e107245 (2021).
  * Sonnen and Janda [2021] K. F. Sonnen and C. Y. Janda, Biochemical Journal 478, 4045 (2021).
  * Casani-Galdon and Garcia-Ojalvo [2022] P. Casani-Galdon and J. Garcia-Ojalvo, Current Opinion in Cell Biology 78, 102130 (2022).
  * Purvis and Lahav [2013] J. E. Purvis and G. Lahav, Cell 152, 945 (2013).
  * Pérez-Verdugo  _et al._ [2023] F. Pérez-Verdugo, S. Banks, and S. Banerjee, arXiv preprint arXiv:2310.04950 (2023).
  * Rombouts  _et al._ [2023] J. Rombouts, J. Elliott, and A. Erzberger, EMBO reports , e57739 (2023).
  * Di Talia and Vergassola [2022] S. Di Talia and M. Vergassola, Annual review of biophysics 51, 327 (2022).
  * Corominas-Murtra and Petridou [2021] B. Corominas-Murtra and N. I. Petridou, Frontiers in Physics 9, 666916 (2021).
  * Guirao and Bellaïche [2017] B. Guirao and Y. Bellaïche, Current opinion in cell biology 48, 113 (2017).
  * Hausberg and Röger [2018a] S. Hausberg and M. Röger, Nonlinear Differential Equations and Applications NoDEA 25, 17 (2018a).
  * Brauns  _et al._ [2020] F. Brauns, J. Halatek, and E. Frey, Physical Review X 10, 041036 (2020).
  * Ross  _et al._ [2021] A. B. Ross, J. D. Langer, and M. Jovanovic, Molecular & Cellular Proteomics 20 (2021).
  * Hausberg and Röger [2018b] S. Hausberg and M. Röger, Nonlinear Differential Equations and Applications NoDEA 25, 1 (2018b).
  * Milo and Phillips [2015] R. Milo and R. Phillips,  _Cell biology by the numbers_ (Garland Science, 2015).
  * Sankaran  _et al._ [2009] J. Sankaran, M. Manna, L. Guo, R. Kraut, and T. Wohland, Biophysical journal 97, 2630 (2009).
  * Shamir  _et al._ [2016] M. Shamir, Y. Bar-On, R. Phillips, and R. Milo, Cell 164, 1302 (2016).
  * Buccitelli and Selbach [2020] C. Buccitelli and M. Selbach, Nature Reviews Genetics 21, 630 (2020).
  * Jacobson  _et al._ [2019] K. Jacobson, P. Liu, and B. C. Lagerholm, Cell 177, 806 (2019).
  * Trimble and Grinstein [2015] W. S. Trimble and S. Grinstein, Journal of Cell Biology 208, 259 (2015).
  * Wyatt  _et al._ [2016] T. Wyatt, B. Baum, and G. Charras, Current opinion in cell biology 38, 68 (2016).
  * Tran-Son-Tay  _et al._ [1991] R. Tran-Son-Tay, D. Needham, A. Yeung, and R. Hochmuth, Biophysical journal 60, 856 (1991).
  * Maître and Heisenberg [2011] J.-L. Maître and C.-P. Heisenberg, Current opinion in cell biology 23, 508 (2011).
  * Chu  _et al._ [2004] Y.-S. Chu, W. A. Thomas, O. Eder, F. Pincet, E. Perez, J. P. Thiery, and S. Dufour, The Journal of cell biology 167, 1183 (2004).
  * Ridgway  _et al._ [2023] W. J. M. Ridgway, M. P. Dalwadi, P. Pearce, and S. J. Chapman, Phys. Rev. Lett. 131, 228302 (2023).
  * van der Kolk  _et al._ [2023] J. van der Kolk, F. Raßhofer, R. Swiderski, A. Haldar, A. Basu, and E. Frey, Phys. Rev. Lett. 131, 088201 (2023).
  * Wadhams and Armitage [2004] G. H. Wadhams and J. P. Armitage, Nature reviews Molecular cell biology 5, 1024 (2004).
  * Antunes and de Souza [2016] G. Antunes and F. M. S. de Souza, Methods in cell biology 132, 127 (2016).
  * Murphy  _et al._ [2022] K. Murphy, C. Weaver, L. Berg, G. Barton, and C. Janeway,  _Janeway’s Immunobiology_ , Vol. 10th ed. (W.W. Norton and Company, New York, NY, 2022, 2022).
  * Alberts  _et al._ [2022c] B. Alberts, R. Heald, A. Johnson, D. Morgan, M. Raff, K. Roberts, and P. Walter,  _Molecular Biology of the Cell_ (WW Norton & Company, 2022) Chap. 22.
  * Guisoni  _et al._ [2017] N. Guisoni, R. Martinez-Corral, J. Garcia-Ojalvo, and J. de Navascués, Development 144, 1177 (2017).
  * Rusilowicz-Jones  _et al._ [2022] E. V. Rusilowicz-Jones, S. Urbé, and M. J. Clague, Molecular Cell (2022).
  * Beckstead  _et al._ [2009] B. L. Beckstead, J. C. Tung, K. J. Liang, Z. Tavakkol, M. L. Usui, J. E. Olerud, and C. M. Giachelli, Journal of Biomedical Materials Research Part A: An Official Journal of The Society for Biomaterials, The Japanese Society for Biomaterials, and The Australian Society for Biomaterials and the Korean Society for Biomaterials 91, 436 (2009).
  * Santillán [2008] M. Santillán, Mathematical Modelling of Natural Phenomena 3, 85 (2008).
  * Olsauskas-Kuprys  _et al._ [2013] R. Olsauskas-Kuprys, A. Zlobin, and C. Osipo, OncoTargets and therapy , 943 (2013).
  * Ilagan  _et al._ [2011] M. X. G. Ilagan, S. Lim, M. Fulbright, D. Piwnica-Worms, and R. Kopan, Science signaling 4, rs7 (2011).
  * Fryer  _et al._ [2004] C. J. Fryer, J. B. White, and K. A. Jones, Molecular cell 16, 509 (2004).
  * Tveriakhina  _et al._ [2023] L. Tveriakhina, G. Scanavachi, E. D. Egan, R. B. D. C. Correia, A. P. Martin, J. M. Rogers, J. S. Yodh, J. C. Aster, T. Kirchhausen, and S. C. Blacklow, bioRxiv 10.1101/2023.09.27.559780 (2023).
  * Cohen  _et al._ [2023] R. Cohen, S. Taiber, O. Loza, S. Kasirer, S. Woland, and D. Sprinzak, Science Advances 9, eadd2157 (2023).
  * Cohen  _et al._ [2022] R. Cohen, S. Taiber, O. Loza, S. Kasirer, S. Woland, and D. Sprinzak, Mechano-signaling feedback underlies precise inner hair cell patterning in the organ of corti, bioRxiv (2022), doi: 10.1101/2019.12.11.123456.
  * Fischer-Friedrich  _et al._ [2014] E. Fischer-Friedrich, A. A. Hyman, F. Jülicher, D. J. Müller, and J. Helenius, Scientific reports 4, 6213 (2014).
  * Seib and Klein [2021] E. Seib and T. Klein, Biology of the Cell 113, 401 (2021).
  * Vázquez-Ulloa  _et al._ [2022] E. Vázquez-Ulloa, K.-L. Lin, M. Lizano, and C. Sahlgren, Critical Reviews in Biochemistry and Molecular Biology 57, 377 (2022).
  * Kandachar and Roegiers [2012] V. Kandachar and F. Roegiers, Current opinion in cell biology 24, 534 (2012).
  * Bray [2016] S. J. Bray, Nature reviews Molecular cell biology 17, 722 (2016).
  * Maître  _et al._ [2016] J.-L. Maître, H. Turlier, R. Illukkumbura, B. Eismann, R. Niwayama, F. Nédélec, and T. Hiiragi, Nature 536, 344 (2016).
  * Dhooge  _et al._ [2008] A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer, and B. Sautois, Mathematical and Computer Modelling of Dynamical Systems 14, 147 (2008).

## Supplementary Material

###  .1 Microscopic dynamics of signaling and adhesion

In the following, we consider a specific set of microscopic processes that can
give rise to Eqs. (3)–(5). We derive the dynamics in the bulk $\Omega$ and at
the surface $\Gamma$ of molecules controlling (i) adhesion at contact
surfaces, and (ii) the exchange of chemical signals, motivated by the
biochemical processes within and at the surface of biological cells. We
consider continuity equations for the particle densities within the bulk $c$
and at the surface $m$ of the form [56]

$\displaystyle\partial_{t}c$
$\displaystyle=D_{\mathrm{c}}\nabla^{2}c+\mathcal{R}_{\mathrm{c}}$ (6)
$\displaystyle\partial_{t}m$
$\displaystyle=D_{\mathrm{m}}\nabla^{2}m+\mathcal{R}_{\mathrm{m}}$ (7)

in which we consider a diffusive flux with coefficients
$D_{\rm{c}},D_{\rm{m}}$ in three and two dimensions respectively, and reaction
terms $\mathcal{R}_{\rm{c}}$ and $\mathcal{R}_{\rm{m}}$. We do not consider
convective flows or other active transport processes here. Bulk and surface
densities are coupled by the boundary condition

$-D_{\mathrm{c}}(\mathbf{n}\cdot\nabla)\left.c\right|_{\Gamma}=j$ (8)

in which $j$ is the flux between bulk and surface and $\mathbf{n}$ is the
normal vector to the surface pointing outwards. A simple form of this flux is
given by [57, 43]

$j=\left.k_{\mathrm{on}}c\right|_{\Gamma}-k_{\mathrm{off}}m$ (9)

with $k_{\mathrm{on}}$ setting the rate with which molecules bind to the
surface and $k_{\mathrm{off}}$ the rate with which they are released from the
surface into the bulk. Note that a positive flux indicates that more molecules
bind to the surface than are released into the bulk. Reactions in the bulk
follow [58]

$\displaystyle\mathcal{R}_{\mathrm{c}}=k_{\rm{p}}-k_{\rm{d}}c$ (10)

with $k_{\mathrm{p}}$ describing an active production of molecules (e.g. due
to protein translation in cells) and $k_{\mathrm{d}}$ their rate of decay
(protein degradation). The bulk production of molecules drives the system out
of thermodynamic equilibrium. For the surface reactions
$\mathcal{R}_{\mathrm{m}}$, we consider different molecular processes
governing adhesion and contact-based signaling, as specified in the following
sections.

Averaging Eq. (6) over the bulk’s volume $V$ and using Eq. (8) and (10) yields
the dynamic equation for the average bulk concentration $\langle c\rangle$

$\displaystyle\frac{d\langle
c\rangle}{dt}=k_{\mathrm{p}}-k_{\mathrm{d}}\langle
c\rangle-\frac{1}{V}\int_{\Gamma}jdA.$ (11)

We define the steady state average density in the absence of boundary flux as
the reference density $c^{0}=k_{\mathrm{p}}/k_{\mathrm{d}}$, and defining
$m^{0}=k_{\mathrm{on}}c^{0}/k_{\mathrm{off}}$ permits introducing normalized
particle densities $c/c^{\mathrm{0}}$ and $m/m^{\mathrm{0}}$.

With diffusion timescale $\tau_{\mathrm{D}}=V^{2/3}/D_{\mathrm{c}}$ and
reaction timescales $\tau_{\mathrm{R}}=1/k_{\mathrm{d}}$ and
$\tau_{\mathrm{on}}=V^{1/3}/k_{\mathrm{on}}$, Eq. (6) and Eq. (8) with time
rescaled in units of $\tau=t/\tau_{\rm{R}}$ read

$\displaystyle\frac{\tau_{\mathrm{D}}}{\tau_{\rm{R}}}\frac{\partial_{\tau}c}{c^{0}}$
$\displaystyle=V^{2/3}\nabla^{2}\frac{c}{c^{0}}+\frac{\tau_{\mathrm{D}}}{\tau_{\mathrm{R}}}\left(1-\frac{c}{c^{0}}\right),$
(12)
$\displaystyle\left(\mathbf{n}\cdot\nabla\right)\left.\frac{c}{c^{0}}\right|_{\Gamma}$
$\displaystyle=\frac{\tau_{\mathrm{D}}}{\tau_{\mathrm{on}}V^{1/3}}\left(\frac{m}{m^{0}}-\frac{c}{c^{0}}\right).$
(13)

In the the following, we consider the limit in which bulk diffusion is much
faster than the reaction kinetics, i.e.
$\tau_{\mathrm{D}}\ll\tau_{\mathrm{R}}$ and
$\tau_{\mathrm{D}}\ll\tau_{\mathrm{on}}$. In this limit, the boundary
condition Eq. (13) is reflective and Eq. 12 becomes a Laplace equation that is
solved by a uniform concentration set by the solution of Eq. (11) (_shadow
limit_ [59]). Inside a cell, a typical diffusion constant for a monomeric
protein is about $10\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{2}\text{\,}{\mathrm{s}}^{-1}$, i.e. it takes roughly 10 seconds for a
protein to traverse a eukaryotic cell [60]. Coefficients of lateral diffusion
on cellular membranes are variable and on the order of
$0.011\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}\text{\,}{\mathrm{s}}^{-1}$
[43, 61]. Most biochemical reactions are catalyzed by enzymes and occur within
less than a second [62], however transcription and translation, i.e. the
synthesis of new proteins, and protein degradation take minutes to hours and
can vary greatly between different proteins [63, 62].

The surface of an adherent droplet consists of two regions separated by a
contact line: the domain of the contact interface $\Gamma_{\rm{c}}$ and the
outer surface $\Gamma_{\rm{f}}$ [Fig. 1(a)]. At the outer surface, molecules
can be exchanged with the bulk, but no adhesion or signaling dynamics occur
within the surface, thus, the reaction term is
$\left.\mathcal{R}_{m}\right|_{\Gamma_{\mathrm{f}}}=j$. We assume in the
following that the contact line separating the two surfaces forms a diffusive
barrier, i.e. that no molecules can diffuse laterally between the surfaces.
This assumption substantially simplifies our calculations, and indeed
diffusion barriers based on protein structures associated with the membrane,
the lipid composition or extreme curvatures—as given at the contact line—have
been found to impede diffusive transport on cellular membranes [64, 65]. The
boundary conditions for the surface densities on the two domains then are

$\displaystyle\left.(\mathbf{n}\cdot\nabla)m\right|_{\partial\Gamma_{\mathrm{f}}}=0$
(14)
$\displaystyle\left.(\mathbf{n}\cdot\nabla)m\right|_{\partial\Gamma_{\mathrm{c}}}=0$
(15)

in which $\partial\Gamma_{\mathrm{f}},\partial\Gamma_{\mathrm{c}}$ denote the
contact line. From Eqs. (7) follows at steady-state
$\left.j\right|_{\Gamma_{\mathrm{f}}}=0$ and the uniform steady-state bulk
concentration

$c=\frac{k_{\mathrm{p}}}{k_{\mathrm{d}}}-\dfrac{1}{k_{\mathrm{d}}V}{\displaystyle\int_{\Gamma_{\mathrm{c}}}}jdA$
(16)

only depends on processes in the bulk and at the contact site
$\Gamma_{\mathrm{c}}$.

In the following sections, we analyse the distribution of molecules regulating
signaling and adhesion in the limit of fast bulk diffusion. In particular, we
introduce the reaction terms and corresponding boundary fluxes for the
adhesion and signaling dynamics at the contact site and compute the steady-
state bulk and surface densities that fulfill Eqs. (7), (16), and boundary
condition $\eqref{eq:bc_Ac}$.

###  .2 Microscopic dynamics of adhesion

Figure 4: Schematic of adhesion dynamics for (a) a single adherent cell on a
substrate and (b) adhesion complexes forming at a cell-cell interface. In the
bulk, adhesion molecules (N) are produced (transcription + translation of
genes) with a rate depending on the regulator concentration $c_{\mathrm{U}}$
and they decay over time with a constant rate. Adhesion molecules are
exchanged between bulk and surface. At the cell-cell interface (b), two
adhesion molecules—one from each cell—bind across the interface and form an
adhesion complex. The maximum surface density of adhesion molecules is set by
the number of available binding sites $m_{\mathrm{N}}^{\mathrm{max}}$ (a) and
$m_{\mathrm{NN}}^{\mathrm{max}}$ (b), respectively. $c_{\mathrm{N}}$ indicates
bulk concentrations, $m_{\mathrm{N}}$ (a) and $m_{\mathrm{NN}}$ (b) the
surface densities of adhesion complexes and $k$ the various rate parameters.

We consider droplets containing adhesion molecules N with a bulk concentration
$c_{\mathrm{N}}$, in contact either with an external substrate, or with
another droplet. At a droplet-substrate interface, these molecules can adhere
to the substrate with a surface density $m_{\mathrm{N}}$ [Fig. 4(a)], whereas
at a droplet-droplet interface, molecules from the two droplets bind to each
other and form complexes with surface concentration $m_{\mathrm{NN}}$ [Fig.
4(b)]. Cells for example produce integrin and cadherin molecules that form
transmembrane complexes to adhere to external structures or other cells
respectively [34].

We assume that the contact areas are given by the configuration minimizing the
droplet surface energy (e.g. Eq. 1), i.e. that shape relaxation dynamics are
fast compared to changes in adhesion density. For cells and cellular
aggregates, the viscoelastic response to mechanical stresses comprises active
and passive components and exhibits relaxation time scales spanning different
orders of magnitudes [66]. For instance, when neutrophil cells are deformed by
micropipette aspiration and subsequently expelled from the pipette, it takes
around a minute for the cell to recover its spherical shape and shape
evolution is well described by a Newtonian liquid drop with a constant
cortical tension approaching its equilibrium configuration [67]. In contrast,
protein production and degradation take place on a timescale of tens of
minutes to several hours [63, 62].

####  .2.1 Adherent droplet on a substrate

We begin with a droplet that forms a physical contact with a solid substrate,
to which adhesion molecules can bind [Fig. 4(a)]. A mass-action based reaction
term for the surface concentration Eq. (7) reads

$\mathcal{R}_{m_{\mathrm{N}}}=k_{\mathrm{on}}^{\mathrm{N}}(m_{\mathrm{N}}^{\mathrm{max}}-m_{\mathrm{N}})c_{\mathrm{N}}-k_{\mathrm{off}}^{\mathrm{N}}m_{\mathrm{N}}$
(17)

with $m_{\mathrm{N}}^{\mathrm{max}}$ the density of available binding sites at
the contact. The flux coupling bulk and contact surface is
$j_{\mathrm{N}}=\mathcal{R}_{m_{\mathrm{N}}}$ and adhesion molecules bound to
the substrate are fixed in place, i.e. $D_{m_{\mathrm{N}}}=0$ in Eq. (7). At
steady-state, it follows from Eqs. (7), (16),(17) and boundary condition Eq.
(15) that $j_{\mathrm{N}}=0$,
$c_{\mathrm{N}}=k_{\mathrm{p}}^{\mathrm{N}}/k_{\mathrm{d}}^{\mathrm{N}}$, and

$\displaystyle
m_{\mathrm{N}}=\dfrac{k_{\mathrm{on}}^{\mathrm{N}}k_{\rm{p}}^{\mathrm{N}}}{k_{\mathrm{on}}^{\mathrm{N}}k_{\rm{p}}^{\mathrm{N}}+k_{\mathrm{off}}^{\mathrm{N}}k_{\rm{d}}^{\rm{N}}}m_{\mathrm{N}}^{\mathrm{max}}.$
(18)

The same expression can also be derived from the grand canonical ensemble.

At steady-state, the flux coupling bulk and surface concentrations [Eq. (8)]
vanishes and the surface can be considered to be in chemical and thermal
equilibrium with a constant temperature $T$ and in contact with a bath of
constant chemical potential $\mu=\mu(c_{\mathrm{N}})$ set by the steady-state
bulk concentration. Note that the chemical potential is kept constant through
a non-equilibrium process – the turnover of adhesion molecules. Each binding
site at the interface is a two-state system: a binding site is either occupied
or unoccupied. If $n$ is the number of occupied binding sites,
$n^{\mathrm{max}}$ the total number of available binding sites at the surface
and $\epsilon$ the binding energy, then the grand canonical partition sum for
the whole surface reads

$\displaystyle\Xi=\sum_{n=0}^{n^{\mathrm{max}}}\binom{n^{\mathrm{max}}}{n}e^{\beta
n(\mu-\epsilon)}=\left(1+e^{\beta(\mu-\epsilon)}\right)^{n^{\mathrm{max}}}$
(19)

with $\beta=(k_{\mathrm{B}}T)^{-1}$. The ensemble average of the number of
occupied binding sites is

$\displaystyle\langle
n\rangle=\frac{1}{\beta}\frac{\partial\ln\Xi}{\partial\mu}=\frac{n^{\mathrm{max}}}{1+e^{\beta(\epsilon-\mu)}},$
(20)

showing that the system follows Fermi-Dirac statistics. In the chemical
equilibrium, the rates of binding and unbinding must be equal for each binding
site. The binding rate of adhesion molecules is

$\displaystyle
k_{\mathrm{binding}}=k_{\mathrm{on}}^{\mathrm{N}}c_{\mathrm{N}}p_{\mathrm{uoc}}$
(21)

with $p_{\mathrm{uoc}}=1/(1+e^{\beta(\mu-\epsilon)})$ the probability that a
binding site is not occupied, while the unbinding rate is

$\displaystyle
k_{\mathrm{unbinding}}=k_{\mathrm{off}}^{\mathrm{N}}p_{\mathrm{oc}}$ (22)

with $p_{\mathrm{oc}}=e^{\beta(\mu-\epsilon)}/(1+e^{\beta(\mu-\epsilon)})$ the
probability that a binding site is occupied. From
$k_{\mathrm{binding}}=k_{\mathrm{unbinding}}$ and Eqs. (21)–(22) follows

$\displaystyle\frac{k_{\mathrm{off}}^{\mathrm{N}}}{k_{\mathrm{on}}^{\mathrm{N}}c_{\mathrm{N}}}=e^{\beta(\epsilon-\mu)}.$
(23)

From $m_{\mathrm{N}}=\langle n\rangle/A_{\mathrm{c}}$ and
$m_{\mathrm{N}}^{\mathrm{max}}=n^{\mathrm{max}}/A_{\mathrm{c}}$ together with
Eq. (20) and $c_{\rm{N}}=k_{\rm{p}}^{\rm{N}}/k_{\rm{d}}^{\rm{N}}$ follows then
Eq. (18).

Expansion in the dilute limit
$k_{\mathrm{on}}^{\mathrm{N}}k_{\rm{p}}^{\rm{N}}/k_{\rm{d}}^{\rm{N}}\ll
k_{\mathrm{off}}^{\mathrm{N}}$, i.e. where saturation effects do not play a
role, yields

$\displaystyle
m_{\mathrm{N}}=\frac{k_{\mathrm{on}}^{\mathrm{N}}k_{\rm{p}}^{\mathrm{N}}}{k_{\mathrm{off}}^{\mathrm{N}}k_{\rm{d}}^{\rm{N}}}m_{\mathrm{N}}^{\mathrm{max}}+\mathcal{O}\left(\left(\frac{k_{\mathrm{on}}^{\mathrm{N}}k_{\rm{p}}^{\rm{N}}}{k_{\mathrm{off}}^{\mathrm{N}}k_{\rm{d}}^{\rm{N}}}\right)^{2}\right).$
(24)

Given that each adhesion complex reduces the surface energy by $\epsilon$
[68], the surface tension at the contact site in this limit is

$\gamma_{\mathrm{c}}=\gamma_{0}-\epsilon\frac{k_{\mathrm{on}}^{\mathrm{N}}k_{\rm{p}}^{\rm{N}}}{k_{\mathrm{off}}^{\mathrm{N}}k_{\rm{d}}^{\rm{N}}}m_{\mathrm{N}}^{\mathrm{max}},$
(25)

with $\gamma_{0}$ the baseline tension containing all other contributions to
the interfacial tension.

####  .2.2 Droplet-droplet adhesion

At contact surfaces between two droplets [Fig. 4(b)], adhesion molecules
produced within the droplets can bind across the interface and form adhesion
complexes with surface density $m_{\mathrm{NN}}$ [36]. Taking exclusion
effects into account, adhesion complexes can only form at unoccupied sites on
the interface. The density of unoccupied sites is
$(m_{\mathrm{NN}}^{\mathrm{max}}-m_{\mathrm{NN}})$ with
$m_{\mathrm{NN}}^{\mathrm{max}}$ the maximum possible density of adhesion
complexes. The reaction term for the density of adhesion complexes is then

$\displaystyle\mathcal{R}_{m_{\mathrm{NN}}}=k_{\mathrm{on}}^{\mathrm{NN}}(m_{\mathrm{NN}}^{\mathrm{max}}-m_{\mathrm{NN}})c_{\mathrm{N},1}c_{\mathrm{N},2}-k_{\mathrm{off}}^{\mathrm{NN}}m_{\mathrm{NN}}$
(26)

with indices $\\{1,2\\}$ labeling the two droplets. The flux coupling bulk and
surface densities is $j_{\mathrm{NN}}=\mathcal{R}_{m_{\mathrm{NN}}}$, and the
tension at the droplet-droplet interface in the dilute limit
$k_{\mathrm{on}}^{\mathrm{NN}}(k_{\rm{p}}^{\rm{N}}/k_{\rm{d}}^{\rm{N}})^{2}\ll
k_{\mathrm{off}}^{\mathrm{NN}}$ is

$\gamma_{\mathrm{c}}=\gamma_{0}-\epsilon\frac{k_{\mathrm{on}}^{\mathrm{NN}}}{k_{\mathrm{off}}^{\mathrm{NN}}}\left(\frac{k_{\rm{p}}^{\rm{N}}}{k_{\rm{d}}^{\rm{N}}}\right)^{2}m_{\mathrm{NN}}^{\mathrm{max}}.$
(27)

Indeed, the force necessary to separate two adhesive cells has been shown to
scale linearly with the squared total number of adhesion molecules [68, 69].
Note that in general the production or decay rates of adhesion molecules in
the bulk [Eq. (10)] can differ between the two cells (Sec. .3.2). In
particular, we consider in the following that the production rate of adhesion
molecules $k_{\mathrm{p}}^{\mathrm{N}}$ depends on the cell-intrinsic
signaling state (Secs. .3.1 and .3.2).

###  .3 Microscopic dynamics of biochemical signaling interactions at contact
surfaces

Figure 5: Schematic of signaling reaction kinetics for (a) a single adherent
cell on a ligand-covered substrate and (b) signals exchanged at a cell-cell
contact. Receptor (R, gray) and ligand (L, green) molecules bind across the
interface and form receptor-ligand complexes (RL). In the single cell, ligands
are substrate-bound whereas receptors are produced in the bulk and exchange
with the surface. For cell pairs, we consider both receptors and ligands to
bind and unbind from the cell-cell interface. Receptor-ligand complexes are
cleaved irreversibly, which produces a signaling molecule (S, gray) that is
released into the bulk. Regulator molecules (U, green) are produced with a
rate that depends on the bulk concentration of signal molecules. The regulator
U in turn determines the production rate of new ligands. In the single cell on
the substrate (a), the ligand molecule is released upon the cleavage event and
can bind a new receptor molecule. In the cell pair, we assume that the
remaining part of the RL complex is degraded after the cleavage event. $c$
indicates bulk concentrations, $m$ surface densities and $k$ the various rate
parameters.

Cells respond to molecular signals from the environment by changing their
internal properties. Many cellular signals are transmitted via the binding of
chemicals to receptor molecules located at the cell surface. These chemical
events trigger internal processes, which result in changes to the molecular
composition, spatial organisation, and corresponding functions of cells [11].
Signaling processes are relevant for example in chemotactic bacteria [70, 9,
71, 72], sensory neurons of the olfactory system [73], immune cell activation
by pathogenic molecules [74] or stem cells responding to differentiation cues
[75].

Signals can arise at physical contacts with surrounding structures (e.g. the
extracellular matrix) or neighboring cells such that the geometry of contacts
in a given system determines the possible signaling partners and affects the
signaling outcome [14, 76, 44, 43]. For instance, in the Notch signaling
pathway, signaling relies on direct interactions between membrane-bound
receptor and ligand molecules [Fig. 5]. Their interaction triggers a series of
proteolytic cleavage events of the receptor at the end of which the
intracellular receptor domain dissociates from the surface and translocates
into the nucleus, where it regulates the expression of target genes [15].
Following the example of the Notch pathway, we consider the reaction-diffusion
dynamics of receptors (R), ligands (L), and receptor-ligand complexes (RL) at
a signaling interface, and derive how the bulk concentration of signal
molecules (S) responds to these interactions at the boundary [Fig. 5]. Similar
to how the Notch intracellular domain regulates gene expression, we consider
that the concentration of signal molecules in turn determines the production
rate of a regulator molecule (U) in the bulk (e.g. a cellular transcription
factor) that controls the production rates of adhesion and signaling
molecules. Given that the regulation of transcription, translation and the
turnover of proteins requires tens of minutes to hours and can vary greatly
between different protein species [62, 77], we assume that the timescale
associated with the regulator turnover $\tau_{\mathrm{u}}$ dominates the
feedback dynamics. On this timescale, we assume that bulk and surface
concentrations relax to their steady state solutions, and that the shape takes
on the corresponding equilibrium configuration. In cells, concentration and
shape dynamics are indeed typically at least an order of magnitude faster –
set by diffusive, biochemical, and viscoelastic timescales which are on the
order of seconds to minutes [66, 43, 61, 62].

####  .3.1 A cell on a signal transmitting substrate

We begin with a single cell in contact with a solid substrate that is covered
with ligands at a fixed uniform density $m_{\mathrm{L}}^{\mathrm{max}}$,
similar to experimental systems developed for the Notch pathway in _in vitro_
assays [78] [Fig. 5(a)]. The cell contains receptor molecules, signaling
molecules, and regulator molecules with bulk concentrations
$c_{\mathrm{R}},c_{\mathrm{S}},$ and $c_{\mathrm{U}}$ respectively, whose
dynamics are coupled via the reactions at the contact surface. We do not
explicitly consider a bulk concentration of ligands, because the substrate has
no receptor molecules to bind to—the cell is only receiving, but not sending
signals. To describe the signaling dynamics at the surface, we use Eq. (7) for
the surface densities of receptors $m_{\rm{R}}$, substrate-bound ligands
$m_{\rm{L}}$ and receptor-ligand complexes $m_{\rm{RL}}$ with the reaction
terms adapted from Khait et al. (2016) [43]

$\displaystyle\mathcal{R}_{m_{\mathrm{R}}}$
$\displaystyle=k_{\rm{on}}^{\rm{R}}c_{\rm{R}}-(k_{\rm{off}}^{\rm{R}}+k_{+}m_{\rm{L}})m_{\rm{R}}+k_{-}m_{\rm{RL}},$
(28) $\displaystyle\mathcal{R}_{m_{\mathrm{L}}}$
$\displaystyle=(k_{-}+k_{\rm{s}})m_{\rm{RL}}-k_{+}m_{\rm{L}}m_{\rm{R}},$ (29)
$\displaystyle\mathcal{R}_{m_{\mathrm{RL}}}$
$\displaystyle=k_{+}m_{\rm{L}}m_{\rm{R}}-(k_{-}+k_{\rm{s}})m_{\rm{RL}},$ (30)

which are explained in the following [Fig. 5(a)]. Receptors are recruited to
the surface with a rate set by $k^{\rm{R}}_{\rm{on}}$ (exocytosis) and they
are removed from the surface with rate $k^{\rm{R}}_{\rm{off}}$ (endocytosis).
Receptors at the contact surface bind ligands at a rate determined by $k_{+}$
to form receptor-ligand complexes, which unbind with rate $k_{-}$. Receptor-
ligand complexes undergo an irreversible enzymatic cleavage with rate
$k_{\rm{s}}$ upon which a fragment of the bound receptor molecule is released
into the bulk and acts as a signaling molecule (S), the remaining part is
degraded, and the ligand is released within the surface where it can bind to a
new receptor molecule.

The bulk concentration $c_{\mathrm{S}}$ of signaling molecules controls the
bulk production rate $k_{\mathrm{p}}^{\mathrm{U}}(c_{\mathrm{S}})$ of the
regulator U [see below, Eq. (41)]. The bulk concentrations of receptors
$c_{\mathrm{R}}$ and signaling molecules $c_{\mathrm{S}}$ are coupled to the
signaling dynamics at the contact via Eq. (16).

We consider the ligands on the substrate to be bound and fixed in place, such
that $D_{m_{\mathrm{L}}}=0,D_{m_{\mathrm{RL}}}=0$ in Eq. (7). The density of
unbound ligands is the difference between the total density of ligands
covering the substrate and the density of receptor-ligand complexes
$m_{\mathrm{L}}=m_{\mathrm{L}}^{\mathrm{max}}-m_{\mathrm{RL}}$. Equations (7)
and (30) together with this relation permit expressing the steady-state
concentration of receptor-ligand complexes in terms of the steady-state
receptor concentration as

$\displaystyle m_{\mathrm{RL}}$
$\displaystyle=\frac{m_{\mathrm{R}}}{m_{\mathrm{R}}+\dfrac{k_{\mathrm{s}}+k_{-}}{k_{+}}}m_{\mathrm{L}}^{\mathrm{max}}.$
(31)

Given Eqs. (7), (28) and (31), the steady-state relation for the distribution
of receptors reads

$0=D_{m_{\mathrm{R}}}\nabla^{2}m_{\rm{R}}+k_{\rm{on}}^{\rm{R}}c_{\mathrm{R}}-k_{\rm{off}}^{\rm{R}}m_{\rm{R}}-\frac{k_{s}k_{+}m_{\mathrm{R}}}{k_{+}m_{\mathrm{R}}+k_{\mathrm{s}}+k_{-}}m_{\mathrm{L}}^{\mathrm{max}}.$
(32)

In the limit where the rate of receptors binding to ligands on the substrate
is large compared to the transport of receptors from the surface into the
bulk, i.e.

$\displaystyle k_{+}m_{\mathrm{L}}^{\mathrm{max}}\gg
k_{\mathrm{off}}^{\mathrm{R}},$ (33)

solutions of Eq. (32) are uniform and follow

$\displaystyle
m_{\rm{R}}=\frac{k_{\rm{on}}^{\rm{R}}c_{\rm{R}}(k_{-}+k_{\rm{s}})}{k_{+}(k_{\rm{s}}m_{\rm{L}}^{\rm{max}}-k_{\rm{on}}^{\rm{R}}c_{\rm{R}})},$
(34)

under boundary condition Eq. (15). The bulk and surface densities of receptors
are coupled via the flux [Eq. (16)]

$\displaystyle j_{\mathrm{R}}$
$\displaystyle=k_{\mathrm{on}}^{\mathrm{R}}c_{\mathrm{R}}-k_{\mathrm{off}}^{\mathrm{R}}m_{\mathrm{R}}.$
(35)

Using Eq. (34) and assuming (33), the flux can be approximated as
$j_{\rm{R}}=k_{\rm{on}}^{\rm{R}}c_{\rm{R}}$ and the steady-state bulk and
surface densities of receptors that follow from Eqs. (16) and (34) are given
by

$\displaystyle c_{\rm{R}}$
$\displaystyle=\frac{k_{\rm{p}}^{\rm{R}}V}{k_{\rm{d}}^{\rm{R}}V+k_{\rm{on}}^{\rm{R}}A_{\rm{c}}},$
(36) $\displaystyle m_{\rm{R}}$
$\displaystyle=\frac{k_{\rm{on}}^{\rm{R}}k_{\rm{p}}^{\rm{R}}(k_{-}+k_{\rm{s}})V}{k_{+}[k_{\rm{s}}m_{\rm{L}}^{\rm{max}}(k_{\rm{on}}^{\rm{R}}A_{\rm{c}}+k_{\rm{d}}^{\rm{R}}V)-k_{\rm{on}}^{\rm{R}}k_{\rm{p}}^{\rm{R}}V]}.$
(37)

For Notch receptors, reported values are
$k_{+}=$0.167\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$ and
$k_{\mathrm{off}}^{\rm{R}}=$0.02\text{\,}{\mathrm{s}}^{-1}$$ [43] [Tab.] and
Notch activation assays with cells on ligand-coated substrates are performed
with surface densities of up to
$m_{\rm{L}}^{\rm{max}}\approx${10}^{5}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-2}$$ [78], which justifies limit (33) and allows to neglect the
$k_{\rm{off}}^{\rm{R}}$-term in Eq. (32). Importantly, $m_{\rm{R}}$ and
$c_{\rm{R}}$ have upper bounds: in the absence of ligands
($m_{\mathrm{L}}^{\mathrm{max}}=0$), the steady-state receptor density is
uniform at
$m_{\rm{R}}^{0}=k_{\mathrm{on}}^{\mathrm{R}}c_{\mathrm{R}}^{0}/k_{\mathrm{off}}^{\mathrm{R}}$
with bulk concentration
$c_{\rm{R}}^{0}=k_{\mathrm{p}}^{\mathrm{R}}/k_{\mathrm{d}}^{\mathrm{R}}$.
Because receptors are removed upon receptor-ligand binding and subsequent
cleavage, $m_{\rm{R}}^{0}$ and $c_{\mathrm{R}}^{0}$ are upper bounds to the
steady-state concentrations. If we estimate the term in brackets of Eq. (32)
using $m_{\rm{R}}=m_{\rm{R}}^{0}$ and typical parameter values as listed in
Tab. 1, neglecting $k_{\rm{off}}^{\rm{R}}$ is valid if
$m_{\rm{L}}^{\rm{max}}\gg$10\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-2}$$.

Given Eqs. (31) and (37), the steady-state density of receptor-ligand
complexes is

$\displaystyle
m_{\rm{RL}}=\frac{k_{\rm{on}}^{\rm{R}}k_{\rm{p}}^{\rm{R}}V}{k_{\rm{s}}(k_{\rm{on}}^{\rm{R}}A_{\rm{c}}+k_{\rm{d}}^{\rm{R}}V)}.$
(38)

The bulk concentration of signaling molecules follows Eq. (16), without a bulk
production term ($k_{\mathrm{p}}^{\mathrm{S}}=0$) and with flux
$j_{\mathrm{S}}=-k_{\mathrm{s}}m_{\mathrm{RL}}$, arising from the cleavage of
receptor-ligand molecules at the surface. The steady-state bulk concentration
is given by

$\displaystyle
c_{\mathrm{S}}=\frac{k_{\rm{on}}^{\rm{R}}k_{\rm{p}}^{\rm{R}}A_{\mathrm{c}}}{k_{\rm{d}}^{\rm{S}}(k_{\rm{on}}^{\rm{R}}A_{\rm{c}}+k_{\rm{d}}^{\rm{R}}V)}.$
(39)

In general, the concentration of signaling molecules depends non-linearly on
the size of the contact area. When the number of receptors that are recruited
to the surface and lost in the signaling process is small compared to the
turnover of molecules in the bulk, we can expand the bulk concentration of
signal molecules

$\displaystyle
c_{\mathrm{S}}=\frac{k_{\rm{p}}^{\rm{R}}k_{\rm{on}}^{\rm{R}}A_{\rm{c}}}{k_{\rm{d}}^{\rm{S}}k_{\rm{d}}^{\rm{R}}V}+\mathcal{O}\left(\left(\frac{k_{\mathrm{on}}^{\mathrm{R}}A_{\mathrm{c}}}{k_{\mathrm{d}}^{\mathrm{R}}V}\right)^{2}\right).$
(40)

In _in vitro_ experiments, a roughly linear relation between Notch response
and contact area was measured also for large contact areas [43].

The production of regulator molecules U in the bulk depends on the steady-
state concentration of signal molecules
$k_{\mathrm{p}}^{\mathrm{U}}(c_{\mathrm{s}})$ —the more signal molecules are
present, the more regulator molecules are produced. The regulation of genes
and the synthesis of new proteins involve multiple steps and molecular
species, which leads to the presence of nonlinear effects like cooperative
binding and multimerization, commonly captured using Hill-functions [79].
Similar to previous studies modeling canonical Notch signaling [16, 42, 30,
41], we therefore assume that steady-state solutions of $c_{\mathrm{U}}$ are
bounded within a concentration range $c_{\mathrm{U}}^{\mathrm{min}}\leq
c_{\mathrm{U}}\leq c_{\mathrm{U}}^{\mathrm{max}}$ [Fig. 6] and we consider a
nonlinear production rate with Hill coefficient $h$

$\displaystyle
k_{\mathrm{p}}^{\mathrm{U}}(c_{\mathrm{S}})=\frac{1}{\tau_{\mathrm{u}}}\left(c_{\mathrm{U}}^{\mathrm{min}}+\frac{(c_{\mathrm{U}}^{\mathrm{max}}-c_{\mathrm{U}}^{\mathrm{min}})}{1+\left(\frac{c_{\mathrm{S}}^{\mathrm{crit}}}{c_{\mathrm{S}}}\right)^{h}}\right)$
(41)

in which $\tau_{\mathrm{u}}=1/k_{\mathrm{d}}^{\mathrm{U}}$ is the
characteristic time scale on which $c_{\mathrm{U}}$ is changing, and
$c_{\mathrm{S}}^{\mathrm{crit}}$ is the critical concentration at the
inflection point [Fig. 6].

The normalized signal $s=c_{\mathrm{S}}/c_{\mathrm{S}}^{\mathrm{crit}}$ can be
written using Eq. (39) as

$\displaystyle s=\chi\frac{A_{\mathrm{c}}}{A_{0}},$ (42)

in which we introduced the _signal susceptibility_

$\displaystyle\chi=\frac{k_{\mathrm{p}}^{\rm{R}}k_{\rm{on}}^{\rm{R}}}{c_{\mathrm{S}}^{\mathrm{crit}}k_{\mathrm{d}}^{\mathrm{S}}k_{\rm{d}}^{\rm{R}}V}A_{\mathrm{0}}.$
(43)

using the definition of the volume-dependent reference area
$A_{0}=(3V/2)^{2/3}\pi^{1/3}$. The volume-dependence of the susceptibility
arises because the degradation of molecules in the bulk scales with the
volume, and due to the reference area $A_{0}$, yielding a scaling of
$\chi\propto V^{-1/3}$. However, in cells where protein degradation does not
increase with the cell volume, the signal susceptibility might increase with
volume. Interestingly, the signal susceptibility is independent of the
cleavage rate $k_{\rm{s}}$. A common experimental perturbation to Notch
signaling is the pharmacological inhibition of the enzyme cleaving the
receptor-ligand complexes (treatment of cells with $\gamma$-secretase
inhibitors) [80]. Our result suggests that the signal susceptibility and thus
the steady-state concentration of signaling molecules is independent of
$k_{\rm{s}}$ unless cleavage is completely prevented. We can estimate the
order of magnitude of the susceptibility using
$k_{\rm{on}}^{\rm{R}}k_{\rm{p}}^{\rm{R}}/k_{\rm{d}}^{\rm{R}}=$2\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-2}\text{\,}{\mathrm{s}}^{-1}$$ [43],
$V=$500\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{3}$$,
$k_{\rm{d}}^{\rm{S}}=$5\text{\times}{10}^{-3}\text{\,}{\mathrm{min}}^{-1}$$
[81, 82] and $c_{\rm{S}}^{\rm{crit}}=1000/V$ [83] yielding $\chi\sim 3000$.

The uniform bulk concentration $c_{\mathrm{U}}$ evolves on the slowest
timescale of the system, the regulatory time scale $\tau_{\mathrm{u}}$,
according to Eq. (11), in which $j_{\mathrm{U}}=0$ because regulator molecules
do not bind to the surface. The saturating response to the received signal
[Eq. (41)] permits introducing a dimensionless _signaling state_ variable

$\displaystyle
u:=\frac{c_{\mathrm{U}}-c_{\mathrm{U}}^{\mathrm{min}}}{c_{\mathrm{U}}^{\mathrm{max}}-c_{\mathrm{U}}^{\mathrm{min}}},$
(44)

normalized to the response range such that $u\in[0,1]$. Eqs. (11), (41), (42),
and (44) lead to the following dynamical equation for the signaling state

$\displaystyle\tau_{\mathrm{u}}\frac{du}{dt}=\sigma(s)-u,$ (45)

with sigmoidal response function

$\displaystyle\sigma(s)=\frac{s^{h}}{1+s^{h}}.$ (46)

as given in the main text.

Figure 6: Sigmoid signal response function Eq. (41) describing the production
rate of regulator molecules $k_{\mathrm{p}}^{\mathrm{U}}$ on time scale
$\tau_{\mathrm{u}}$ within the concentration range
$c_{\mathrm{U}}^{\mathrm{min}}\leq c_{\mathrm{U}}\leq
c_{\mathrm{U}}^{\mathrm{max}}$ depending on the signal molecule concentration
$c_{\mathrm{S}}$. For the temporal dynamics of the normalized signaling state
variable $u$ [Eq. (44)], this can be mapped to a corresponding response
function $\sigma$ [Eq. (46)] scaling between 0 and 1 [see also Fig. 1(c)].
Plot shown for $h=4$ (blue) and in the limit $h\rightarrow\infty$ (gray),
where the response is a step function of $c_{\rm{S}}$.

##### Signal-dependent active mechanics.—

In many biological systems, for instance mechanosensory epithelia [84, 85,
16], adhesion molecules are expressed downstream of contact-based signals.
Accordingly, we consider that the production rate of adhesion molecules
$k_{\mathrm{p}}^{\mathrm{N}}(c_{\mathrm{U}})$ is a monotonously increasing
function of the regulator concentration. As stated above, such a function is
generally expected to be nonlinear, e.g. due to cooperativity between
molecules involved in protein synthesis [79]. In the following, we assume that
$k_{\mathrm{p}}^{\mathrm{N}}$ vanishes for $c_{\mathrm{U}}\leq
c_{\mathrm{U}}^{\mathrm{min}}$, i.e. no adhesion molecules are produced when
the regulator concentration drops below $c_{\mathrm{U}}^{\mathrm{min}}$, and
we linearize $k_{\mathrm{p}}^{\mathrm{N}}$ around
$c_{\mathrm{U}}^{\mathrm{min}}$

$\displaystyle
k_{\mathrm{p}}^{\mathrm{N}}(c_{\mathrm{U}})=\left.\frac{dk_{\mathrm{p}}^{\mathrm{N}}}{dc_{\mathrm{U}}}\right\rvert_{c_{\mathrm{U}}^{\mathrm{min}}}(c_{\mathrm{U}}-c_{\mathrm{U}}^{\mathrm{min}})+\mathcal{O}\left((c_{\mathrm{U}}-c_{\mathrm{U}}^{\mathrm{min}})^{2}\right).$
(47)

With Eq. (44) the surface tension at the contact site [Eq. (25)] can then be
written as

$\displaystyle\gamma_{\mathrm{c}}=\gamma_{0}-\gamma_{\mathrm{A}}u$ (48)

with the _adaptive adhesion coefficient_

$\displaystyle\gamma_{\mathrm{A}}=\epsilon\frac{k_{\mathrm{on}}^{\mathrm{N}}(c_{\mathrm{U}}^{\mathrm{max}}-c_{\mathrm{U}}^{\mathrm{min}})}{k_{\mathrm{off}}^{\mathrm{N}}k_{\mathrm{d}}^{\mathrm{N}}}\left.\frac{dk_{\mathrm{p}}^{\mathrm{N}}}{dc_{\mathrm{U}}}\right\rvert_{c_{\mathrm{U}}^{\mathrm{min}}}m_{\mathrm{L}}^{\mathrm{max}}.$
(49)

We can estimate the magnitude of $\gamma_{\rm{A}}$ for a cell that detaches if
$u=0$ (i.e. $A_{\rm{c}}=0$ and
$\gamma_{\rm{c}}=\gamma_{\rm{m}}$+$\gamma_{\rm{f}}$) and acquires a half-
sphere shape if $u=1$ (i.e. $A_{\rm{c}}=A_{0}$ and
$\gamma_{\rm{c}}=\gamma_{\rm{m}}$), in which case
$\gamma_{\rm{A}}=\gamma_{\rm{f}}$. In cells, $\gamma_{\rm{f}}$ is typically on
the order of $1\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$ [86].

##### Feedback between contact-based signaling and adaptive mechanics creates
bistability.—

Equations (42) and (48) describe respectively how transmitted signals depend
on the area of the cell-substrate interface, and how the interfacial tension
in turn depends on the signaling state. Given that cell shape dynamics are
fast compared to the signaling timescale $\tau_{\rm{u}}$, the area is
determined quasi-instantaneously by the conjugate interfacial tension
$\gamma_{\rm{c}}$.

Figure 7: a) Schematic of a spherical-cap-shaped droplet attached to a
substrate. The droplet has an outer surface area $A_{\mathrm{f}}$ and forms a
contact of size $A_{\mathrm{c}}$ with the substrate. $\gamma_{\mathrm{c}}$ and
$\gamma_{\mathrm{f}}$ are the area-associated surface tensions and
$\gamma_{\mathrm{m}}$ the interfacial tension between substrate and
surrounding medium. b) Normalized contact area $A_{\mathrm{c}}/A_{0}$ at
minimal surface energy (Eq. 51) as a function of the surface tension ratio
$(\gamma_{\mathrm{c}}-\gamma_{\mathrm{m}})/\gamma_{\mathrm{f}}$.

We consider the equilibrium configuration that minimizes the surface energy of
an incompressible droplet in contact with a solid substrate

$\displaystyle
E=\int_{A_{\mathrm{c}}}(\gamma_{\mathrm{c}}-\gamma_{\mathrm{m}})dA+\int_{A_{\mathrm{f}}}\gamma_{\mathrm{f}}dA+E_{0}$
(50)

with $\gamma_{\mathrm{m}}$ the energy per area between the substrate and the
surrounding medium and $E_{0}$ depending on the total size of the substrate
[Fig. 7(a)]. The surface energy is minimal when the droplet takes the shape of
a spherical cap with normalized contact area [Fig. 7(b)]

$\displaystyle\frac{A_{\mathrm{c}}}{A_{0}}=\left[1-\left(\dfrac{\gamma_{\mathrm{c}}-\gamma_{\mathrm{m}}}{\gamma_{\mathrm{f}}}\right)^{2}\right]\left[\frac{2}{(2-\dfrac{\gamma_{\mathrm{c}}-\gamma_{\mathrm{m}}}{\gamma_{\mathrm{f}}})(1+\dfrac{\gamma_{\mathrm{c}}-\gamma_{\mathrm{m}}}{\gamma_{\mathrm{f}}})^{2}}\right]^{2/3}.$
(51)

Equations (42), (45), (48), and (51) describe the dynamics of the signaling
state $u$ and contact area $A_{\mathrm{c}}$ of the adherent cell. We use a
combination of linear stability analysis and simulations to study the steady-
state solutions of this system. For different parameter combinations of $\chi$
and $\gamma_{\mathrm{A}}$, either one or two stable steady-state solutions
$u^{*}$ are found [Fig.(2)(c)]. In particular, a bistable regime appears above
a critical value of the adaptive tension $\gamma_{A}^{\mathrm{cusp}}$, bounded
by two saddle-node bifurcation lines (SN), which converge in a cusp node
[Fig.(2)(c)]. For $\gamma_{\rm{A}}>\gamma_{A}^{\mathrm{cusp}}$ and small
$\chi$, the only stable solution is a configuration with small contact area
$A_{\mathrm{c}}$, correspondingly weak signal transmission and a low signaling
state $u$. For $\chi>\chi_{\rm{SN1}}$ —above the lower SN line— a second
stable configuration appears with a large contact area $A_{\mathrm{c}}$, which
permits a stronger signaling interaction with the substrate and a larger
signaling state $u$ [Fig.(2)(c), inset]. This latter configuration is
accessible only when the positive feedback between signaling and adaptive
mechanics is sufficiently strong. The same bistability between a small and a
large contact area configurations is also the basis for the shape oscillations
we report in pairs of adaptive droplets [Fig. 2 (d)].

In the limit $h\rightarrow\infty$ [Eq. (46)], i.e. when the production of
regulator molecules responds to signals in a step-wise manner [Fig. 6], one
can derive a simple relation between $\chi$ and $\gamma_{A}$ for the two
saddle-node lines. In general in this limit, the only possible stable steady-
state solutions of Eq. (45) are $u^{*}\in\\{0;1\\}$ and the corresponding
surface tensions at the contact site [Eq. (48)] are
$\gamma_{c}\in\\{\gamma_{0};\gamma_{0}-\gamma_{A}\\}$. For small values of
$\chi$, signaling is weak and the only stable steady-state is $u^{*}=0$ with a
small contact area set by $\gamma_{c}=\gamma_{0}$. The second stable steady-
state $u^{*}=1$ appears for

$\displaystyle s(\left.A_{\mathrm{c}}\right|_{\gamma_{0}-\gamma_{A}})\geq 1.$
(52)

For

$\displaystyle s(\left.A_{\mathrm{c}}\right|_{\gamma_{0}})>1,$ (53)

the configuration with small contact area and $u=0$ is no longer a steady-
state solution and $u^{*}=1$ remains the only stable steady-state. From
conditions (52)–(53) together with (42) follows that the critical
susceptibilities at the saddle-node lines delineating the bistable regime are
given by

$\displaystyle\chi_{1}=\frac{A_{0}}{\left.A_{\mathrm{c}}\right|_{\gamma_{\mathrm{c}}=\gamma_{0}-\gamma_{\mathrm{A}}}}$
(54)

and

$\displaystyle\chi_{2}=\frac{A_{0}}{\left.A_{\mathrm{c}}\right|_{\gamma_{\mathrm{c}}=\gamma_{0}}}.$
(55)

with $\chi_{1}$ describing the lower and $\chi_{2}$ the upper saddle-node line
[Fig. 8(a)]. Note that in the discussed limit the upper saddle-node
bifurcation line ($\chi_{2}$) is independent of the adaptive adhesion
coefficient $\gamma_{\mathrm{A}}$ [Fig. 8(a)]. In general, the saddle-node
bifurcation lines and therefore the size of the bistable regime depend on the
baseline tension $\gamma_{0}$. Fig. 8(b) shows for the discussed limit how the
size of the bistable regime increases with the baseline tension.

Figure 8: The state diagram of a single adherent droplet on a signal-
transmitting solid substrate depends on (a) the Hill coefficient $h$ in the
nonlinear response function [Eq.(46)] and (b) the baseline tension
$\gamma_{0}$. Lines are saddle-node bifurcation lines that converge in a cusp
node bifurcation (filled circles) and that encompass a bistable regime
[compare to Fig. 2 (b)]. The reference susceptibilities are (a)
$\chi_{0}=A_{0}/\left.A_{\mathrm{c}}\right|_{h=\infty}$ and (b)
$\chi_{0}=A_{0}/\left.A_{\mathrm{c}}\right|_{(\gamma_{0}-\gamma_{m})/\gamma_{f}=0.9}$.
(a) $(\gamma_{0}-\gamma_{m})/\gamma_{f}=0.95$, (b) $h=\infty$.

####  .3.2 Signaling in pairs of cells

In this section, we consider contact-dependent signals exchanged between two
cells indexed with $i,j\in\\{1,2\\}$ that share an interface [Fig. 5(b)]. In
addition to containing signaling and regulator molecules, each cell produces
receptors as well as ligands, which exchange between bulk and surface –
ligands are not substrate-bound with fixed positions as we considered in the
preceding part. The receptors on the surface of cell $i$ bind to the ligands
on the surface of the other cell $j$ and vice versa, producing respectively
oriented receptor-ligand complexes. Upon cleavage they release signal
molecules into the receptor-carrying cell $i$. Contrary to the way we treated
substrate-bound ligands in the preceding section, ligands at the droplet
interface are not released after cleavage of the receptor-ligand complexes,
but are degraded together with the remaining receptor fragment instead [87].
While some literature suggests that ligands can also be recycled after a
signaling event or enter alternative signaling pathways [88, 89], we here
consider that receptors and ligands are always degraded after cleavage.
Signaling molecules control the production of regulator molecules as before,
which feed back onto the production terms. In line with the typical molecular
mechanisms in Notch signaling, we consider an active regulation of _ligand_
production [90]. As explained in Sec. .1, the steady-state bulk concentrations
of receptor, ligand, signaling, and regulator molecules are uniform within
each cell with a value set by the flux balance condition [Eq. (16)]. To
capture the reaction-diffusion dynamics at the interface, we study Eq. (7) for
receptors, ligands, and complexes using the reaction terms [43]

$\displaystyle\mathcal{R}_{\mathrm{R}}$
$\displaystyle=k_{\mathrm{on}}^{\mathrm{R}}c_{\mathrm{R},i}-(k_{\mathrm{off}}^{\mathrm{R}}+k_{+}m_{\mathrm{L},j})m_{\mathrm{R},i}+k_{-}m_{\mathrm{RL},i}$
(56) $\displaystyle\mathcal{R}_{\mathrm{L}}$
$\displaystyle=k_{\mathrm{on}}^{L}c_{\mathrm{L},j}-(k_{\mathrm{off}}^{\mathrm{L}}+k_{+}m_{\mathrm{R},i})m_{\mathrm{L},j}+k_{-}m_{\mathrm{RL},i}$
(57) $\displaystyle\mathcal{R}_{\mathrm{RL}}$
$\displaystyle=k_{+}m_{\mathrm{R},i}m_{\mathrm{L},j}-(k_{-}+k_{\mathrm{s}})m_{\mathrm{RL},i},$
(58)

with rate constants as described in Sec. .3.1. Under boundary condition Eq.
(15), steady-state solutions of Eq. (7) for the densities of receptors,
ligands and receptor-ligand complexes with the reaction terms Eqs. (56)-(58)
are uniform and follow the relations

$\displaystyle m_{\mathrm{R},i}$
$\displaystyle=\frac{k_{\rm{on}}^{\rm{R}}c_{\mathrm{R},i}(k_{\rm{s}}+k_{-})}{k_{\rm{off}}^{\rm{R}}(k_{\rm{s}}+k_{-})+k_{+}k_{\rm{s}}m_{\mathrm{L},j}},$
(59) $\displaystyle m_{\mathrm{L},j}$
$\displaystyle=\frac{k_{\rm{on}}^{\rm{L}}c_{\mathrm{L},j}(k_{\rm{s}}+k_{-})}{k_{\rm{off}}^{\rm{L}}(k_{\rm{s}}+k_{-})+k_{+}k_{\rm{s}}m_{\mathrm{R},i}}.$
(60)

Ligand molecules are only produced in the bulk, but not at the surface, thus,
the steady-state concentrations of bulk and surface densities have the upper
limits
$c_{\rm{L}}^{\mathrm{0}}=k_{\mathrm{p}}^{\rm{L}}/k_{\mathrm{d}}^{\rm{L}}$ and
$m_{\rm{L}}^{\mathrm{0}}=k_{\mathrm{on}}^{\rm{L}}c^{\mathrm{0}}/k_{\mathrm{off}}^{\rm{L}}$.
Together with Eqs. (59) and (16) for the receptor bulk concentration, one can
define a lower limit for the surface density of receptors

$\displaystyle
m_{\rm{R}}^{\rm{min}}=\frac{k_{\rm{on}}^{\rm{R}}k_{\rm{off}}^{\rm{L}}(k_{-}+k_{\rm{s}})k_{\rm{p}}^{\rm{R}}k_{\rm{d}}^{\rm{L}}V}{k_{\rm{on}}^{\rm{L}}k_{+}k_{\rm{s}}k_{\rm{p}}^{\rm{L}}(k_{\rm{on}}^{\rm{R}}A_{\rm{c}}+k_{\rm{d}}^{\rm{R}}V)+k_{\rm{d}}^{\rm{L}}k_{\rm{off}}^{\rm{R}}(k_{-}+k_{\rm{s}})k_{\rm{d}}^{\rm{R}}V}.$
(61)

In line with Khait et al. 2016 [43], we consider that cells produce an excess
of receptors compared to the number of ligands, i.e. $k_{\rm{p}}^{\rm{R}}\gg
k_{\rm{p}}^{\rm{L}}$, and similar to limit (33) for the single cell we assume
that the endocytosis rate of ligands is small compared to the rate of binding

$\displaystyle k_{+}m_{\mathrm{R}}^{\rm{min}}\gg k_{\rm{off}}^{\rm{L}},$ (62)

allowing to neglect the $k_{\rm{off}}^{\rm{L}}$-term in Eq. (57). The bulk and
surface densities of ligands are coupled in Eq. (16) via the flux

$\displaystyle
j_{\mathrm{L},j}=k_{\mathrm{on}}^{\mathrm{L}}c_{\mathrm{L},j}-k_{\mathrm{off}}^{\mathrm{L}}m_{\mathrm{L},j},$
(63)

which using Eq. (60) and assuming (62) can be written as
$j_{\rm{L},j}=k_{\rm{on}}^{\rm{L}}c_{\rm{L},j}$. In the described limit,
solving Eqs. (7) with the reaction terms Eqs. (56)–(58) under boundary
condition Eq. (15) together with Eq. (16) and boundary flux Eq. (35) for the
bulk density of receptors yields

$\displaystyle c_{\mathrm{R},i}$
$\displaystyle=\dfrac{k_{\rm{p}}^{\rm{R}}(k_{\rm{on}}^{\rm{L}}A_{\rm{c}}+k_{\rm{d}}^{\rm{L}}V)-k_{\rm{p}}^{\rm{L}}(c_{\mathrm{U},j})k_{\rm{on}}^{\rm{L}}A_{\rm{c}}}{k_{\rm{d}}^{\rm{R}}(k_{\rm{on}}^{\rm{L}}A_{\rm{c}}+k_{\rm{d}}^{\rm{L}}V)},$
(64) $\displaystyle c_{\mathrm{L},j}$
$\displaystyle=\frac{k_{\rm{p}}^{\rm{L}}V}{k_{\rm{on}}^{\rm{L}}A_{\rm{c}}+k_{\rm{d}}^{\rm{L}}V},$
(65) $\displaystyle m_{\mathrm{R},i}$
$\displaystyle=\frac{A_{\rm{c}}k_{\rm{on}}^{\rm{L}}k_{\rm{on}}^{\rm{R}}(k_{\rm{p}}^{\rm{R}}-k_{\rm{p}}^{\rm{L}}(c_{\mathrm{U},j}))+(k_{\rm{d}}^{\rm{L}}k_{\rm{on}}^{\rm{R}}k_{\rm{p}}^{\rm{R}}-k_{\rm{d}}^{\rm{R}}k_{\rm{on}}^{\rm{L}}k_{\rm{p}}^{\rm{L}}(c_{\mathrm{U},j}))V}{k_{\rm{d}}^{\rm{R}}k_{\rm{off}}^{\rm{R}}(k_{\rm{on}}^{\rm{L}}A_{\rm{c}}+k_{\rm{d}}^{\rm{L}}V)},$
(66) $\displaystyle m_{\mathrm{L},j}$
$\displaystyle=\frac{k_{\rm{on}}^{\rm{L}}k_{\rm{p}}^{\rm{L}}(c_{\mathrm{U},j})k_{\rm{d}}^{\rm{R}}k_{\rm{off}}^{\rm{R}}(k_{-}+k_{\rm{s}})V}{k_{+}k_{\rm{s}}(A_{\rm{c}}k_{\rm{on}}^{\rm{L}}k_{\rm{on}}^{\rm{R}}(k_{\rm{p}}^{\rm{R}}-k_{\rm{p}}^{\rm{L}}(c_{\mathrm{U},j}))+(k_{\rm{d}}^{\rm{L}}k_{\rm{on}}^{\rm{R}}k_{\rm{p}}^{\rm{R}}-k_{\rm{d}}^{\rm{R}}k_{\rm{on}}^{\rm{L}}k_{\rm{p}}^{\rm{L}}(c_{\mathrm{U},j}))V)},$
(67) $\displaystyle m_{\mathrm{RL},i}$
$\displaystyle=\frac{k_{\rm{on}}^{\rm{L}}k_{\rm{p}}^{\rm{L}}(c_{\mathrm{U},j})V}{k_{\rm{s}}(k_{\rm{on}}^{\rm{L}}A_{\rm{c}}+k_{\rm{d}}^{\rm{L}}V)}$
(68)

and the steady-state bulk concentration of signaling molecules following Eq.
(16) with $j_{\mathrm{S}}=-k_{\mathrm{s}}m_{\mathrm{RL}}$ and
$k_{\mathrm{p}}^{\mathrm{S}}=0$ as before is

$\displaystyle
c_{\mathrm{S},i}=\frac{k_{\rm{on}}^{\rm{L}}k_{\rm{p}}^{\rm{L}}(c_{\mathrm{U},j})A_{\rm{c}}}{k_{\rm{d}}^{\rm{S}}(k_{\rm{on}}^{\rm{L}}A_{\rm{c}}+k_{\rm{d}}^{\rm{L}}V)}.$
(69)

We introduced here that the bulk production rate of ligands depends on the
regulator concentration $k_{\mathrm{p}}^{\mathrm{L}}(c_{\mathrm{U}})$. As a
consequence, receptor, ligand and complex concentrations differ according to
the respective regulator concentrations, which evolve on a slower timescale
than the reaction-diffusion dynamics. The dynamic equation for the normalized
regulator concentration in dependence on exchanged signals is given in the
main text [Eq. 4]. In many contexts, signals in the canonical Notch pathway
are mutually inhibitory: the more signal a cell receives, i.e. the more
ligands bind to the receptors on its surface, the lower its production rate of
ligand molecules [90]. We therefore consider that the production rate of
ligands is a monotonously _decreasing_ function of the regulator concentration
$c_{\mathrm{U}}$. We assume that no ligands are produced at
$c_{\mathrm{U}}=c_{\mathrm{U}}^{\mathrm{max}}$, i.e.
$k_{\mathrm{p}}^{\mathrm{L}}(c_{\mathrm{U}}^{\mathrm{max}})=0$, and we expand
$k_{\mathrm{p}}^{\mathrm{L}}$ to first order around
$c_{\mathrm{U}}^{\rm{max}}$

$\displaystyle
k_{\mathrm{p}}^{\mathrm{L}}(c_{\mathrm{U}})=\left.\frac{dk_{\mathrm{p}}^{\mathrm{L}}}{dc_{\mathrm{U}}}\right|_{c_{\mathrm{U}}^{\mathrm{max}}}(c_{\mathrm{U}}-c_{\mathrm{U}}^{\mathrm{max}})+\mathcal{O}\left((c_{\mathrm{U}}-c_{\mathrm{U}}^{\mathrm{max}})^{2}\right).$
(70)

Using the definition of $u$ [Eq. (44)] it follows that

$\displaystyle
k_{\mathrm{p}}^{\mathrm{L}}(u)=\left(-\left.\frac{dk_{\mathrm{p}}^{\mathrm{L}}}{dc_{\mathrm{U}}}\right|_{c_{\mathrm{U}}^{\mathrm{max}}}\right)(c_{\mathrm{U}}^{\mathrm{max}}-c_{\mathrm{U}}^{\mathrm{min}})(1-u).$
(71)

As before [Eq. (40)], we consider the number of ligands that are recruited to
the surface and lost in the signaling process small compared to the turnover
of molecules in the bulk and expand the bulk concentration of signal molecules

$\displaystyle
c_{\mathrm{S},i}=\frac{k_{\rm{p}}^{\rm{L}}(c_{\mathrm{U},j})k_{\rm{on}}^{\rm{L}}A_{\rm{c}}}{k_{\rm{d}}^{\rm{S}}k_{\rm{d}}^{\rm{L}}V}+\mathcal{O}\left(\left(\frac{k_{\mathrm{on}}^{\mathrm{L}}A_{\mathrm{c}}}{k_{\mathrm{d}}^{\mathrm{L}}V}\right)^{2}\right).$
(72)

in accordance with observations that the Notch signal scales roughly linear
with the size of the contact area, including large contacts [43].

Considering Eqs. (71)–(72) we can write the signal
$s_{ij}=c_{\mathrm{s},i}/c_{\mathrm{s}}^{\mathrm{crit}}$ received by cell $i$
from cell $j$ as

$\displaystyle s_{ij}=\chi\frac{A_{\mathrm{c}}}{A_{0}}(1-u_{j})$ (73)

with the _signal susceptibility_

$\displaystyle\chi=\frac{k_{\rm{on}}^{\rm{L}}A_{\mathrm{0}}(c_{\mathrm{U}}^{\mathrm{max}}-c_{\mathrm{U}}^{\mathrm{min}})}{c_{\mathrm{S}}^{\mathrm{crit}}k_{\mathrm{d}}^{\mathrm{S}}k_{\rm{d}}^{\rm{L}}V}\left(-\left.\frac{dk_{\mathrm{p}}^{\mathrm{L}}}{dc_{\mathrm{U}}}\right|_{c_{\mathrm{U}}^{\mathrm{max}}}\right)$
(74)

This expression of the susceptibility is similar to Eq. (43) for a single
cell, but depends on the production, decay and transport rates of ligands
rather than receptors. For the single cell we consider receptor interactions
with an excess of substrate-bound ligands [Eq. (33)], while at cell-cell
contacts, diffusive ligands of one cell bind to receptors of the other cell
and we assume excess production of receptors compared to ligands [Eq. (62)].

##### Signal-dependent active mechanics at the droplet-droplet interface.—

We consider that the production rate of adhesion molecules
$k_{\mathrm{p}}^{\mathrm{N}}$ is a monotonously increasing function of the
regulator concentration $c_{\mathrm{U}}$ as in Sec. .3.1. Linearizing
$k_{\mathrm{p}}^{\mathrm{N}}$ around $c_{\rm{U}}^{\rm{min}}$ under the same
conditions as described in Sec. .3.1, one can write the surface tension at the
contact site Eq. (27) as

$\displaystyle\gamma_{\mathrm{c}}=\gamma_{0}-\gamma_{\mathrm{A}}u_{1}u_{2}$
(75)

with the _adaptive adhesion coefficient_

$\displaystyle\gamma_{\mathrm{A}}=\epsilon\frac{k_{\mathrm{on}}^{\mathrm{NN}}(c_{\rm{U}}^{\rm{max}}-c_{\rm{U}}^{\rm{min}})^{2}}{k_{\mathrm{off}}^{\mathrm{NN}}(k_{\mathrm{d}}^{\rm{N}})^{2}}\left(\left.\frac{dk_{\mathrm{p}}^{\mathrm{N}}}{dc_{\mathrm{U}}}\right|_{c_{\mathrm{U}}^{\mathrm{max}}}\right)^{2}m_{\mathrm{NN}}^{\mathrm{max}}.$
(76)

This expression is identical to Eq. (49) except for the squared terms arising
from the production and decay of adhesion molecules, because both cells need
to contribute molecules for the formation of adhesion complexes at the
interface. [Fig. 4(b)] The adaptive adhesion coefficient has units of energy
per area–the units of a surface tension–and must be on the same order of
magnitude as $\gamma_{0}$ for the adaptive tension to induce significant shape
changes. In cells, the baseline tension depends on the passive material
properties of the cell membrane and associated protein structures, as well as
additional active processes like the contractile forces created by the
actomyosin cortex. The overall cell surface tension is usually on the order of
$1\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$ [86]. Cellular adhesion
complexes involve several molecular species and their formation not only
depends on the binding of adhesion molecules across the cell-cell interface,
but also the anchoring to internal cytoskeleton, which itself exhibits complex
dynamics and feedback effects [31, 36], thus the energy $\epsilon$ is an
effective energy per adhesion complex reflecting more than just the binding
energy between two adhesion molecules.

##### Symmetry-breaking of signaling states

In many biological systems, Notch signals are mutual inhibitory, i.e. signals
suppress the production of ligands [13]. Strong mutual inhibitory interactions
generically lead to spontaneous symmetry-breaking, whereby small initial
differences in the signaling states are amplified and diverge to high- and
low-value steady states. At the onset of symmetry-breaking, the uniform
steady-state solution of Eq. 4 becomes unstable. To derive an approximation
for the onset of symmetry-breaking, we expand $\sigma(s_{ij})$ [Eq. (46)] for
a general Hill coefficient $h$ to first order around the inflection point
$s_{ij}=1$

$\displaystyle\sigma(s_{ij})=\frac{1}{2}+\frac{h}{4}(s_{ij}-1)+\mathcal{O}((s_{ij}-1)^{2})$
(77)

yielding the dynamic equation

$\displaystyle\tau_{\rm{u}}\frac{du_{i}}{dt}=\frac{1}{2}+\frac{h}{4}(s_{ij}-1)-u_{i}$
(78)

and using the definition of the signal Eq. (5) the uniform steady-state is
[Fig. 9(a)]

$\displaystyle u^{*}=1-\frac{2+h}{4+h\chi\dfrac{A_{\rm{c}}}{A_{0}}}.$ (79)

Linear stability analysis reveals that this uniform steady state looses
stability for

$\displaystyle\chi_{\mathrm{PF}}=\frac{4A_{0}}{hA_{\mathrm{c}}},$ (80)

with $A_{\mathrm{c}}=A_{\mathrm{c}}(\gamma_{\mathrm{c}})$ and
$\gamma_{\mathrm{c}}=\gamma_{0}-\gamma_{\mathrm{A}}(u^{*})^{2}$. Comparison
with the steady-state contact area computed numerically along the
supercritical pitchfork bifurcation line that was derived via continuation in
MatCont shows good agreement [Fig. 9(b)]. Figure. 10 shows the normalized
steady-state contact area $A_{\rm{c}}/A_{0}$ in the state space of feedback
parameters.

Figure 9: (a) Uniform fixpoints of Eq. (4) computed numerically (gray) and
approximation from linearization of the response function $\sigma(s_{ij})$
around $s_{ij}=1$ [Eq. (79)], (orange), $h=4$. (b) Comparison between Eq. (80)
(orange) and the steady-state contact area computed numerically along the
supercritical pitchfork bifurcation line derived via continuation in MatCont
(gray), $\gamma_{0}/2\gamma_{\rm{f}}=0.98$ Figure 10: State diagram as shown
in Fig. 2(a) for $\gamma_{0}/2\gamma_{\rm{f}}=0.98$. The color code indicates
the normalized contact area at the fixpoint, i.e. at the uniform fixpoint
state below the PF and SHET line and at the symmetry-broken fixpoints above
the PF and Hopf line. The oscillatory regime between Hopf and SHET line is
white as it does not contain any stable fixpoints.

###  .4 Shapes of asymmetric droplets

Figure 11:  (a) Parameterization of a pair of asymmetric droplets adapted from
[91]. (b) The droplet volumes can be expressed in terms of the three spherical
cap volumes $v_{1},v_{2},v_{\rm{c}}$ (c) Differences in droplet volumes
$\delta V/\bar{V}=\\{0.25,0.5\\}$ (blue) or outer interfacial tensions
$\delta\gamma_{\rm{f}}/\bar{\gamma_{\rm{f}}}=\\{0.25,0.5\\}$ (brown) change
how the contact area between the droplets depends on the tension ratio
$\gamma_{\rm{c}}/2\bar{\gamma}_{\rm{f}}$

For pairs of droplets with unequal volumes ($V_{1}\neq V_{2}$) or outer
surface tensions ($\gamma_{\rm{f},1}\neq\gamma_{\rm{f},2}$), Eq. (2) does not
describe the size of the contact area. To derive the equilibrium shape and
contact size of asymmetric droplets, we compute the minimum of the surface
energy

$\displaystyle
E=\gamma_{\rm{c}}A_{\rm{c}}+\gamma_{\rm{f},1}A_{\rm{f},1}+\gamma_{\rm{f},2}A_{\rm{f},2}$
(81)

under constant volume constraint. We follow the approach and use the
parameterization introduced in [91], which is shown in Fig. 11(a). The droplet
volumes can be expressed in terms of three spherical cap volumes $v_{i}$ with
$i\in\\{1,2,\rm{c}\\}$ [Fig. 11(b)] such that

$\displaystyle V_{1}=v_{1}+v_{c}$ (82) $\displaystyle V_{2}=v_{2}-v_{c}.$ (83)

Given the radii of curvature $R_{i}$ and the radius $r$ as shown in Fig.
11(a), we can define the length scales

$\displaystyle a_{i}=\sqrt{R_{i}^{2}-r^{2}}$ (84)

and surfaces

$\displaystyle
H_{i}(a_{i},r)=\frac{1}{2}\left(a_{i}^{2}+r^{2}+a_{i}\sqrt{a_{i}^{2}+r^{2}}\right),$
(85)

which allows to express the spherical cap volumes as

$\displaystyle
v_{i}(a_{i},r)=\frac{\pi}{3}\left(a_{i}+\sqrt{a_{i}^{2}+r^{2}}\right)^{2}\left(2\sqrt{a_{i}^{2}+r^{2}}-a_{i}\right)$
(86)

and the different droplet surfaces as

$\displaystyle A_{i}(a_{i},r)=4\pi H_{i}(a_{i},r).$ (87)

Using these definitions and expressing the outer surface tensions as
$\gamma_{\rm{f},1}=\bar{\gamma}_{\rm{f}}+\delta\gamma_{\rm{f}},\gamma_{\rm{f},2}=\bar{\gamma}_{\rm{f}}-\delta\gamma_{\rm{f}}$,
we can rewrite Eq. (81) as

$\displaystyle\frac{E}{4\pi\bar{\gamma}_{\rm{f}}}$
$\displaystyle=\left(1+\frac{\delta\gamma_{\rm{f}}}{\bar{\gamma}_{\rm{f}}}\right)H_{1}(a_{1},r)+\left(1-\frac{\delta\gamma_{\rm{f}}}{\bar{\gamma}_{\rm{f}}}\right)H_{2}(a_{2},r)$
$\displaystyle+2\left(\frac{\gamma_{\rm{c}}}{2\bar{\gamma}_{\rm{f}}}\right)H_{\rm{c}}(a_{\rm{c}},r)$
(88)

with $\bar{\gamma}_{\rm{f}}=(\gamma_{\rm{f},1}+\gamma_{\rm{f},2})/2$. The
minima in terms of the four parameters ($a_{1},a_{2},a_{\rm{c}},r$) under
constant volume constraints $V_{1}=\bar{V}-\delta V,V_{2}=\bar{V}+\delta V$
were computed numerically, allowing to derive the size of the contact area
$A_{\rm{c}}=4\pi H_{c}(a_{c},r)$ (Sec. 2).

###  .5 Literature values for reaction and diffusion rates

Khait et al. 2016 obtained quantitative estimates for many parameters
governing the dynamics of Notch receptors and ligands (Tab. 1, [43]). The
authors measured the 2D diffusion constant $D_{\mathrm{m}}$ and the
endocytosis rate $k_{\rm{off}}$ in different cell lines and estimated the
reaction rates from previously reported measurements of the binding kinetics
for soluble molecules in 3D. The rate of receptor and ligand transport from
the bulk to the surface $k_{\rm{on}}^{\rm{R}}c_{\rm{R}}^{0}$ and
$k_{\rm{on}}^{\rm{L}}c_{\rm{L}}^{0}$ were estimated by assuming that the
steady-state surface densities are
$m_{\rm{R}}^{0}=$100\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-2}$$ and
$m_{\rm{L}}^{0}=$10\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-2}$$ (i.e. there
is an excess of receptors) when a cell is not in contact with another cell or
ligand-coated substrate. In that case
$m_{\rm{R}}=m_{\rm{R}}^{0}=k_{\rm{on}}^{\rm{R}}c_{\rm{R}}^{0}/k_{\rm{off}}^{\rm{R}}$
and
$m_{\rm{L}}=m_{\rm{L}}^{0}=k_{\rm{on}}^{\rm{L}}c_{\rm{L}}^{0}/k_{\rm{off}}^{\rm{L}}$.
$A_{\rm{c}}$ gives the range of contact areas for two spherical cells with
radius $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ that keep a normalized
contact area $A_{\rm{c}}/A_{0}\in[0,1]$ as assumed in this work.

Parameter | Symbol | Value
---|---|---
Endocytosis | $k_{\rm{off}}$ | $0.02\text{\,}{\mathrm{s}}^{-1}$
Cleavage | $k_{\rm{s}}$ | $0.34\text{\,}{\mathrm{s}}^{-1}$
Binding | $k_{+}$ | $0.167\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}\text{\,}{\mathrm{s}}^{-1}$
Unbinding | $k_{-}$ | $0.034\text{\,}{\mathrm{s}}^{-1}$
Diffusion coefficients | $D_{m_{\rm{R}}},D_{m_{\rm{L}}}$ | $0.020.08\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}\text{\,}{\mathrm{s}}^{-1}$
Diffusion coefficient | $D_{m_{\mathrm{RL}}}$ | $(D_{m_{\mathrm{R}}}+D_{m_{\mathrm{L}}})/4$
Exocytosis receptors | $k_{\rm{on}}^{\rm{R}}c_{\rm{R}}^{0}$ | $2\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-2}\text{\,}{\mathrm{s}}^{-1}$
Exocytosis ligands | $k_{\rm{on}}^{\rm{L}}c_{\rm{L}}^{0}$ | $0.2\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-2}\text{\,}{\mathrm{s}}^{-1}$
Contact area | $A_{\rm{c}}$ | $0125\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$
Table 1: Typical parameter values for reaction and diffusion rates of receptor
and ligand molecules as reported in [43] and estimate of common cellular
length scales

###  .6 Numerical methods

Physical quantity |  Symbol |  Values
---|---|---
Base line interfacial tension relative to outer surface tension |  $\gamma_{0}/2\gamma_{\rm{f}}$ |  0.98
Base line interfacial tension relative to outer surface tension in a single adherent droplet |  $(\gamma_{0}-\gamma_{\rm{m}})/2\gamma_{\rm{f}}$ |  0.98
Critical susceptibility without adaptive tension |  $\chi_{0}$ |  40.604
Adaptive adhesion coefficient relative to outer surface tension |  $\gamma_{\rm{A}}/2\gamma_{\rm{f}}$ |  Fig. 2(b): 0.294,
Fig. 2(c): {0.15, 0.21, 0.23, 0.2352, 0.5}
Fig. 2(d): 0.8
Fig. 2(e): {0.4, 0.6, 0.8}
Fig. 3(c): 0.65
Relative signal susceptibility |  $\chi/\chi_{0}$ |  Fig. 2(c): {0.1, 0.61, 0.604, 0.6021, 0.6, 0.95}
Fig. 2(d): {0.4704, 0.7388}
Fig. 3(c): {0.465, 0.5, 0.6, 0.74 }
Volume asymmetry |  $\delta V/\bar{V}$ |  {0.25, 0.5}
Tension asymmetry |  $\delta\gamma_{\rm{f}}/\bar{\gamma}_{\rm{f}}$ |  {0.25, 0.5}
Table 2: Parameter values.

##### Bifurcation analysis

The state and bifurcation diagrams presented in Fig. 2(a),(b), Fig. 3(a),(b)
and Fig. 13 were computed via continuation with the MATLAB-based software
package MatCont [92] (MatCont7p3 and MATLAB R2021a, scripts with details and
numerical settings available at https://git.embl.de/dullwebe/dullweber2024. In
general, initial fixpoints to initialize the continuation were computed by
integration over time with random initial conditions using the Integrator
Method ode45. The saddle-node homoclinic (HSN) and associated non-central
homoclinic to saddle-node (NCH) bifurcations [Fig.3 3(b)] could only be
obtained using the GUI-based version of MatCont7p3. For this, continuation of
limit cycles was performed for $\gamma_{A}/\gamma_{0}=0.95$ and decreasing
continuation parameter $\chi$ until the period reached a value close to 100
indicating the presence of a homoclinic causing period divergence. For
different values of $\gamma_{A}/\gamma_{0}=0.95$, this point consistently
coincides with the saddle-node bifurcation line. From the limit cycle of
largest period, continuation of a HSN was initialized with InitStepsize 0.01,
MinStepsize 0.05, MaxStepsize 0.1, MaxNewtonIters 3, MaxCorrIters 10,
MaxTestIters 10, VarTolerance 1e-6, FunTolerance 1e-6, TestTolerance 1e-5,
Adapt 1, MaxNumPoints 2000, CheckClosed 50 and Jacobian Increment 1e-05.
Continuation of the HSN allows to detect the NCH codimension-2 point.

Results of the continuation were confirmed using simulations and analysis in
Mathematica 13.0 (notebook available at
https://git.embl.de/dullwebe/dullweber2024). Specifically, we tested the
number and types of stable attractors in different parameter regimes with
simulations using NDSolve and ParametricNDSolve with the equation
simplification method _Residuals_. Fixpoints shown in the phase plots Fig.
3(c) were computed numerically in Mathematica from the intersections of
nullclines. The oscillation amplitude [Fig. 2(d)] and period [Fig. 2(a)] were
computed from the extrema of simulated trajectories, and checked against the
dominant Fourier components.

##### Treatment of asymmetric droplet shapes

To obtain estimates of the equilibrium shapes of asymmetric droplets, we
numerically computed the minimum of Eq. (88) in terms of the four parameters
($a_{1},a_{2},a_{\rm{c}},r$) and under the constant volume constraints
$V_{1}=\bar{V}-\delta V,V_{2}=\bar{V}+\delta V$. We computed the contact area
$A_{\rm{c}}=4\pi H_{c}$ for values of $\gamma_{\rm{c}}/2\bar{\gamma}_{\rm{f}}$
evenly spaced on the interval $[0,1]$. For unequal volumes ($V_{1}\neq
V_{2}$), but identical outer surface tensions, we used a 5th order polynomial
to fit a function
$A_{\rm{c}}=A_{\rm{c}}(\gamma_{\rm{c}}/2\bar{\gamma}_{\rm{f}})$ on the
interval $[0,1]$ using Mathematica’s function _Fit_ with the default
_LevenbergMarquardt_ method [Fig. (11)(c)]. For droplets with asymmetric outer
tension, but equal volumes, the droplet with higher outer tension is
completely internalized if
$\gamma_{\rm{c}}/2\bar{\gamma}_{\rm{f}}\leq\delta\gamma_{\rm{f}}$ [91], thus,
we used a piecewise function to fit the contact area with
$A_{\rm{c}}=2^{4/3}A_{0}$ on the interval $[0,\delta\gamma_{\rm{f}}]$. The
interval $[\delta\gamma_{\rm{f}},1]$ was fitted with a combination of a
rational function of the form $a+b/(\gamma_{\rm{c}}/2\gamma_{\rm{f}}-c)^{d}$
close to the threshold of internalisation with fit parameters $a-d$ and a 5th
order polynomial [Fig. 11(c)]. Fits of the contact area were used for
continuation in MatCont and simulations in Mathematica to derive the state
diagrams shown in Fig. 3(a).

All codes are available at https://git.embl.de/dullwebe/dullweber2024.

Figure 12: (a) State diagram as shown in Fig. 2(a) for different values of
$\gamma_{0}/2\gamma_{\rm{f}}=\\{0.98,0.95,0.92,0.9\\}$ (dark to light blue)
showing the saddle-node pitchfork (SP) codimension-2 bifurcation point and
associated bifurcation lines. Note that $\chi_{0}$ is a function of
$\gamma_{0}/2\gamma_{\rm{f}}$. (b) Same state diagram as in (a), but without
normalization of $\chi$ and $\gamma_{\rm{A}}/2\gamma_{\rm{f}}$. Figure 13:
Bifurcation analysis close to the saddle-node pitchfork, Left: Enlarged view
of the state diagram of the doublet in terms of normalized feedback control
parameters shown in Fig. 2(a) close to the saddle-node pitchfork (SP)
codimension-2 bifurcation point. (a) - (d) show stable (solid line) and
unstable (dashed line) fixpoints and saddles (dotted line) computed for
variation of one feedback parameter as indicated by dotted lines in the state
diagram on the left. As the pitchfork interacts with one of the saddle-nodes
(compare (b) and (c)), it changes from supercritical (PF-) to subcritical
(PF+) and the saddle (SN) becomes a Saddle-Heteroclinic (SHET). In the
parameter regime between the H and SHET bifurcation lines, the system has no
stable fixpoints, but stable limit cycles. H: Hopf bifurcation. Diagrams were
computed in MatCont, $\gamma_{0}/2\gamma_{\rm{f}}=0.98$.