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David
Oriola
*^{a},
Miquel
Marin-Riera
^{a},
Kerim
Anlaş
^{a},
Nicola
Gritti
^{a},
Marina
Sanaki-Matsumiya
^{a},
Germaine
Aalderink
^{a},
Miki
Ebisuya
^{a},
James
Sharpe
^{ab} and
Vikas
Trivedi
*^{ac}
^{a}European Molecular Biology Laboratory, EMBL Barcelona, Dr. Aiguader 88, PRBB Building, 08003, Barcelona, Spain. E-mail: david.oriola@embl.es; trivedi@embl.es
^{b}Institució Catalana de Recerca i Estudis Avançats, 08010, Barcelona, Spain
^{c}European Molecular Biology Laboratory, Developmental Biology Unit, Meyerhofstraße 1, 69117 Heidelberg, Germany

Received
12th January 2022
, Accepted 22nd March 2022

First published on 5th May 2022

Multicellular aggregates are known to exhibit liquid-like properties. The fusion process of two cell aggregates is commonly studied as the coalescence of two viscous drops. However, tissues are complex materials and can exhibit viscoelastic behaviour. It is known that elastic effects can prevent the complete fusion of two drops, a phenomenon known as arrested coalescence. Here we study this phenomenon in stem cell aggregates and provide a theoretical framework which agrees with the experiments. In addition, agent-based simulations show that active cell fluctuations can control a solid-to-fluid phase transition, revealing that arrested coalescence can be found in the vicinity of an unjamming transition. By analysing the dynamics of the fusion process and combining it with nanoindentation measurements, we obtain the effective viscosity, shear modulus and surface tension of the aggregates. More generally, our work provides a simple, fast and inexpensive method to characterize the mechanical properties of viscoelastic materials.

Shaping of organs during morphogenesis results from the material response of the constituent tissues to the forces which in turn are generated by them. Understanding the material properties of biological tissues holds the key to elucidating how shape and form emerge during morphogenesis both in vivo during embryonic development,

The mechanical properties of tissues have been measured using a wide range of techniques (for a detailed review see ref. 16 and 17). Absolute measurements of tissue mechanical parameters such as surface tension γ, viscosity η or shear modulus μ, are possible by means of different techniques such as parallel plate compression,^{8,18,19} axisymmetric drop shape analysis,^{9,20} micropipette aspiration^{21} and drop sensors.^{22,23} In all cases, an external force is used to probe the system. A few methods have been used to obtain relative measurements at the tissue scale such as laser ablation^{24} or the fusion of multicellular aggregates.^{25–27} In both cases the measured velocities can be related to material properties. In the first case, the strain rate is related to the ratio of tissue stress σ and viscosity η,^{24} while in the second case the speed of fusion is dictated by the viscocapillary velocity γ/η.^{26,28–31} Of all the previous methods, limited appreciation has been given to the fusion method,^{10,27,32,33} which is arguably one of the simplest methods to obtain relative measures. Additional advantages of the method are the fact that there is no need for a calibrated probe and it is a non-contact method.^{16} The fusion of viscoelastic droplets is known to exhibit a phenomenon known as arrested coalescence,^{34–37} whereby the degree of coalescence is related to the elasticity of the material. The stable anisotropic shapes it can produce, have been exploited extensively to produce emulsions in a wide range of industries like food, cosmetics, petroleum and pharmaceutical formulations.^{34–36,38,39} Interestingly, this phenomenon has also been observed in biological tissues,^{40,41} as well as other active matter systems such as ant^{42} or bacterial^{43} aggregate colonies. Despite the fact that the sintering of drops is a classical problem that has been extensively studied both for passive^{28–30,44–46} and active systems,^{26,31,42,43,47,48} arrested coalescence still remains poorly understood.

In this work, we study the phenomenon of arrested coalescence in stem cell aggregates and show that a minimal Kelvin–Voigt model successfully captures the dynamics of the process. By fitting our model to the fusion dynamics, the viscocapillary velocity v_{c} = γ/η and the shear elastocapillary length _{e} = γ/μ^{49} can be obtained. In addition, we complement these results with nanoindentation measurements to obtain absolute values of the effective viscosity, shear modulus and surface tension of the aggregates. Finally, by using agent-based simulations of the fusion process, we propose a mechanism by which active cell fluctuations can drive a solid-to-fluid phase transition and explore how the supracellular mechanical properties arise from the cell level interactions.

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Fig. 2 (A) Schematics of two identical droplets fusing along the e_{x} axis. θ is the angle of fusion which is π/2 for complete coalescence and takes a value in the range (0, π/2) for arrested coalescence. R(θ) is the radius of each aggregate, r(θ) is the neck radius during the fusion process and L(θ) is the end-to-end length. (B) Time evolution of (r/R)^{2} = sin^{2}θ as a function of the inverse elastocapillary number β for τ = 4 h and ε_{Y} = 0.11 by solving eqn (1) (see Appendix). (C) Bifurcation diagram showing the steady state coalescence angle θ_{max} as a function of β/β_{c}. For β < β_{c} the system undergoes a pitchfork bifurcation where the non-fused state loses stability in favour of the fused state. The numerical steady state solution of eqn (1) is shown as a solid line while the approximate analytical solution assuming R(θ) ≈ R_{0} is shown as a dashed line (see Appendix). ε_{Y} = 0.11. (D) Yield strain dependence of θ_{max} on β by solving eqn (1) numerically at steady state. Inset: β dependence on the yield strain ε_{Y} for different θ_{max} values (in radians). |

Fig. 3 (A) Fusion dynamics quantification showing the averaged time evolution of sin^{2}θ, where θ is the fusion angle of the assembly (see Fig. 2). Three different aggregate sizes are shown (shaded regions denote SD around the mean experimental curve). The numerical fits (solid lines) are obtained using eqn (1). (B) The parameters τ and β scale linearly with the aggregate size as expected from the theory. From the slope the viscocapillary velocity v_{c} and shear elastocapillary length _{e} can be inferred (mean ± SD. Errors in the y-axis are smaller than the symbol size, n = 63 fusion events). |

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Fig. 4 (A) ya‖a agent-based simulations of the fusion of two cell aggregates for F_{p}/K_{adh}r_{max} = 0.20 and α = 1. (B) Averaged time evolution of sin^{2}θ over time (n = 10, shaded region denotes SD around the mean simulation curve) in the simulations for different active fluctuation strengths F_{p}/K_{adh}r_{max} = (0.098, 0.116, 0.134, 0.171, and 0.208). K_{adh}/K_{r} = 1, r_{0}/r_{max} = 0.4, λ/K_{adh}τ_{on} = 1, α = 1, and 500 cells per aggregate. The numerical fits (dashed lines) are obtained using eqn (1). (C) Effect of active fluctuations (AF) to the fusion of the cell aggregates. Color map of logβ in parameter space. Three distinct regions can be identified corresponding to no coalescence, arrested coalescence and complete coalescence. (D) Mean squared relative displacement of cells as a function of time for different active fluctuation strengths F_{p}/K_{adh}r_{max} (same values as in panel B). Cells change from a subdiffusive (∼t^{0.3}) to a diffusive (∼t) behaviour for increasing F_{p}. t_{0}/τ_{on} = 10^{3}. Inset: Viscoelastic relaxation time vs. active fluctuation strength. |

We analyzed the fusion dynamics in the simulations by using the end-to-end length of the assembly as in the experiments and varied the active force F_{p} and the duty ratio α (see Fig. 4). We fitted eqn (1) to the averaged dynamics (Fig. 4B) and extracted the effective macroscopic parameters τ and β. The study revealed the presence of three main regimes depending on β (see Movies S2–S4, ESI†): (i) no coalescence (β ≳ 20), (ii) arrested coalescence (20 ≳ β ≳ 1) and (iii) complete coalescence (β ≲ 1) (Fig. 4C), which qualitatively agree with the regimes found in the continuum model (Fig. 2). The same regimes are also identified when studying the characteristic viscocapillary time τ (see Fig. S4, ESI†). These results suggest that the system undergoes a solid-to-fluid transition for increasing strength or duty ratio of the active fluctuations. To assess if the observed transition is similar to a rigidity or a jamming transition, we studied the relative mean squared displacement of cells in our simulations (Fig. 4D). We found that in regimes (i) and (ii) the behaviour was subdiffusive while the behaviour was mainly diffusive in regime (iii). In addition, we observed that the viscoelastic relaxation time τ_{v} diverges close to the transition point (see Fig. 4D, inset), which is reminiscent of a critical slowing down phenomenon observed in jammed systems.^{53,63,64} In order to verify if phase (i) corresponded to a jammed phase, we performed compression/relaxation cycles in parallel plate compression simulations on the aggregates (see Fig. S5 and Movies S5, S6, ESI†) and identified the presence of a yield stress in regime (i), below which the deformation was not recovered during the relaxation process, indicating a plastic behaviour of the material.^{65} Hence, we conclude that in our simulations, arrested coalescence is found at the vicinity of a solid-to-fluid transition, similarly to jammed systems.

Continuum descriptions of drop coalescence have been mainly limited to purely viscous drops.^{28–31} This has limited the use of such theories to the determination of viscosity and surface tension, despite tissue stiffness and viscoelastic effects having important implications for tissue engineering and being known to play a major role in cancer.^{75,76} Here we present a simple method that when combined with a contact method such as nanoindentation or AFM, allows a fast full mechanical characterisation of 3D tissue aggregates. It is important to notice that nanoindentation measurements are done in a timescale of seconds while fusion experiments occur in a timescale of hours. This is a limitation of our method that should be taken into account when interpreting the measurements. Apart from mouse embryonic stem cells, we successfully applied our model to characterise the arrested coalescence behaviour of human stem cell aggregates (Fig. S6 and Movie S7, ESI†) and human breast epithelial cell aggregates (Fig. S7, based on data from ref. 41). Hence, our method constitutes a promising tool in the bioengineering and medical fields for the mechanical characterization of tissue spheroids. More generally, the method can also be potentially used to characterise the mechanics of inert drops in the emulsion industry. Finally, we envision that future work on the theory of sintering for viscoelastic materials will be important in the formation of biological structures in vitro using bioink units.^{77}

Mouse ES cell culture and 3D aggregate formation.
T/Bra::GFP mouse embryonic stem cells^{78} were maintained in ES-Lif (ESLIF) medium, consisting of KnockOut Dulbecco's Modified Eagle's Medium (DMEM) supplemented with 10% fetal bovine serum (FBS), 1× non-essential aminoacids (NEEA), 50 U mL^{−1} Pen/Strep, 1× GlutaMax, 1× sodium pyruvate, 50 μM 2-mercaptoethanol and leukemia inhibitory factor (LIF). Cells adhered to 0.1% gelatin-coated (Millipore, ES-006-B) tissue culture-treated 25 cm^{2} flasks (T25 flasks, Corning, 353108) in an incubator at 37 °C and 5% CO_{2}. To form the aggregates ∼300 cells were aggregated per well in 96-well U-bottom plates (Greiner Cellstar, #650970) containing 40 μL NDiff227 media (Takara Bio, #Y40002) for 24 h prior to fusion. To ensure the state of the cells was the same in all fusion events, only the multicellular aggregates that did not express T/Bra (mesodermal marker) at 24 h were considered.

Image acquisition, feature extraction and fitting procedure.
2D images of cell aggregates in 96-well microplates were acquired using the high content imaging PerkinElmer Opera Phenix® system in non-confocal bright field mode. A 10× air objective was used with 0.3 N.A. and an exposure time of 100 ms. To capture the dynamics of the fusion process, snapshots were acquired every 10 min for a duration of 10 h. All time points were segmented using the software MOrgAna (Machine-learning based Organoid Analysis),^{50} a Python-based machine learning software (https://github.com/LabTrivedi/MOrgAna.git). To fit our model to the experiments, the end-to-end distance of the assembly L was obtained by fitting an ellipse to the final mask at every time frame. The fitting error due to the irregular shape of the fused assemblies was reduced by averaging over many fusion events (see Fig. S2, ESI†). Finally, using the relationship L(θ) = 2R(θ)(1 + cosθ) and considering L(0) = 4R_{0}, the time evolution of the fusion angle θ(t) was obtained. Finally, the experimental and simulated data were fitted to the solution of eqn (1) using a non-linear least squares method. The solution of eqn (1) was obtained numerically using the Python solver odeint and the fitting was done using curve_fit, both functions from the Python package SciPy.^{79}

Nanoindentation measurements.
The mechanical measurements were done using the Chiaro Nanoindenter (Optics11) adapted to a Leica DMi8 inverted microscope. The aggregates were transferred from the multiwell plates to μ-Slide 8 well coverslips (ibidi, #80826) coated with 0.1% gelatin and containing warm NDiff227. Indentations were done with a spherical cantilever probe of 27 ± 3 μm of radius and a stiffness of 0.025 ± 0.002 N m^{−1}. The approach speed was 5 μm s^{−1} and the indentation depth was ≃6 μm (∼3% of the typical size of an aggregate). The effective elastic modulus E_{eff} was calculated by fitting the Hertz's model to the indentation curves and the corresponding shear modulus was obtained as μ = (E_{eff}/2)(1 − ν^{2})/(1 + ν),^{80} assuming a Poisson's ratio of ν = 1/2. The effective elastic modulus for each aggregate was obtained by averaging around 3–4 measurements. Finally, the average effective elastic modulus was obtained by averaging over n = 25 different aggregates.

Cell aggregate growth.
In the experiments, we find that the radius R of a cell aggregate grows linearly with time. This type of growth behaviour has been observed in other avascular multicellular systems such as tumor spheroids.^{81,82} A model that recapitulates this type of growth considers that only an outer crust of constant thickness d grows with rate Γ, while the rest of the spheroid does not proliferate.^{81} Considering the volume of the spheroid V and the volume of the crust V_{c} we have

We can rewrite the previous equation in terms of the dynamics of the radius:^{81}

For sufficiently long times d/R ≪ 1, and hence the dynamics follows Ṙ ≃ dΓ which leads to

From the last expression we find that, for constant cell density, the dynamics of the cell number N(t) follow

= ΓV_{c} | (3) |

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R(t) ≃ R_{0} + dΓt | (5) |

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and the doubling time will be

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Kelvin–Voigt model.
We consider a multicellular aggregate as a homogeneous incompressible Kelvin–Voigt material drop with effective shear viscosity η, shear modulus μ and surface tension γ. Hereinafter, we use index notation and Einstein's summation convention. The constitutive equation for the stress tensor σ_{ij} reads

where is the symmetric strain tensor, P is the hydrostatic pressure and u_{i} is the displacement field. The continuity equation reads

where v_{i} = _{i}. The latter condition is valid provided that cell proliferation is negligible in the system. Force balance in the bulk in the absence of external forces reads

Similarly, force balance on the surface reads

where H is the local mean curvature of the surface and n_{i} is the unit normal vector to the surface. Let us now consider the fusion of two identical spherical aggregates. The total volume and area of the assembly will be denoted by V and S, respectively. The work of the viscoelastic forces per unit time Ẇ(t) reads

where Γ(t) and ∂Γ(t) denote the integration domains of the volume and surface of the assembly, respectively. Using the force balance in the bulk (eqn (2)), the divergence theorem and force balance on the surface (eqn (11)) one finds^{26,28–31}

where the last equality on the right-hand side corresponds to the work done by the surface tension forces per unit time, which we refer as to Ẇ_{γ}. At the same time, Ẇ_{γ} can also be expressed simply as

Combining eqn (12) and (13) we find^{26,28–31}

The previous expression states that the work done by the bulk forces per unit time equals the work done by the surface forces per unit time. Following previous work,^{26,28–31,45,46} we model the two fusing aggregates as two spherical caps of radius R(θ) with a circular contact ‘neck’ region of radius r(θ) = R(θ)sinθ (see Fig. 2A). The volume V(θ) and surface S(θ) of two fused droplets with a fusion angle θ can be obtained by using simple geometric considerations:^{26}

Given that ∂_{i}v_{i} = 0, the total volume of the assembly V will be conserved during the fusion process. Considering an initial radius of the aggregates R(0) = R_{0}, the total volume of the assembly reads . From the previous equation we obtain R(θ) as follows:^{31}

The dynamics of the fusion process will be completely determined by the evolution of θ(t), with θ(0) = 0 to θ(∞) = θ_{max}. Let us assume the axis of fusion is e_{x} (see Fig. 2A). The end-to-end length L(θ) of the fusion assembly along this axis will be given by

As mentioned in the main text, our hydrodynamic model cannot capture the physics at the onset of fusion and we add a yield strain to effectively account for an elasticity threshold value for the fusion of viscoelastic solid drops.^{36} We consider a shifted rest length L′(0) = L(0) + δL, being δL/L′(0) ≪ 1. We approximate the strain as ∂_{x}u ≃ −ε(θ), where ε(θ) reads

with ε_{Y} = δL/L′(0) and ε_{L}(θ) having the following expression:^{36}

We approximate the corresponding strain rate as ∂_{x}v ≃ −(θ), where (θ) reads

The previous expression differs from the usual one, used in ref. 26 and 28–31, which is based on a definition of strain as changes in the center-to-center length of the fusion assembly (as opposed to end-to-end as in eqn (21)). Using eqn (20) and the fact that the system is incompressible, we obtain an approximation for the strain tensor:

Using the previous simplified expressions we can calculate the work per unit time done by the bulk and surface forces:

where τ = ηR_{0}/γ is the characteristic viscocapillary time and β = μR_{0}/γ is a dimensionless parameter quantifying the degree of fusion. The shear elastocapillary length reads _{e} ≡ γ/μ = R_{0}/β. Using eqn (15) we find an equation for the dynamics of the fusion angle θ(t):

where f(θ), g(θ) read

Notice that from eqn (26) the viscocapillary time τ is a factor of 4 larger than the usual definition.^{26,28–31} This is a consequence of our choice of strain in eqn (21) which is twice as small as the usual one. For small angles and β = 0, eqn (26) reduces to the typical form for the sintering of viscous drops^{30,31} (see eqn (34) in Connection to previous studies of viscous drops). Let us study the stability of the system around θ = 0. The dynamics of a perturbation δθ ≪ 1 reads

Hence for β < β_{c} = 1/ε_{Y}, the non-fused state θ = 0 loses stability. Notice that in the absence of yield strain (i.e. ε_{Y} = 0) the non-fused state is always unstable and hence, two drops in contact will always fuse. The critical condition is equivalent to

which means that the yield point corresponds to when the yield stress σ_{Y} equals the Laplace pressure. When the Laplace pressure is larger than the yield stress of the material fusion starts. Considering R(θ) ≈ R_{0}, we can obtain an analytical expression for θ_{max} as a function of β by solving eqn (26) at steady state:

As expected, the angle of arrested coalescence is independent of the viscocapillary time τ and only depends on β. Close to the critical point β = β_{c}, the arrested fusion angle reads θ_{max} ∼ ε^{1/2}, where ε ≡ (β_{c} − β)/β_{c} is a small parameter that characterizes the distance to the critical point. Hence the system is completely determined by three parameters: τ, β and ε_{Y}.

σ_{ij} = 2η_{ij} + 2με_{ij} − Pδ_{ij} | (8) |

∂_{i}v_{i} = 0 | (9) |

∂_{j}σ_{ij} = 0 | (10) |

σ_{ij}n_{j} = 2γHn_{i} | (11) |

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(13) |

(14) |

Ẇ = Ẇ_{γ} | (15) |

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S(θ) = 4πR^{2}(θ)(1 + cosθ) | (17) |

R(θ) = 2^{2/3}(1 + cosθ)^{−2/3}(2 − cosθ)^{−1/3}R_{0} | (18) |

L(θ) = 2R(θ)(1 + cosθ) | (19) |

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Connection to previous studies of viscous drops.
Here we connect our results to previous classical results in the literature of the sintering of purely viscous droplets. For β = ε_{Y} = 0 and small angles (θ ≪ 1), eqn (26) can be approximated to the well known form:

This expression is equivalent to the typical form with the only difference being that the viscocapillary time is a factor 4 larger than the usual definition.^{77} By solving this equation we find that sin^{2}θ(t) follows^{77}

A well known scaling relation for the evolution of the angle for t ≪ τ is

Another useful expression is the time dependence of the relative shrinkage of the assembly assuming R(t) ≃ R_{0}:

the last expression is related to the evolution of the aspect ratio.^{83,84} For short-timescales (t ≪ τ) the previous expression reduces to

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sin^{2}θ(t) ≃ 1 − e^{−4t/τ} | (33) |

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(35) |

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Active cell–cell interaction dynamics.
At every time step, the active cell–cell dynamics is simulated in the following way: for a set of N cells we define N effective ‘protrusions’, each protrusion P_{i} being produced by each cell i. The protrusion P_{i} always has one end at x_{i} and can be connected to another cell j at x_{j}. At any time point, P_{i} can be “on”, that is connecting cell i to cell j and applying a force of magnitude F_{p} (see main text), or it can be “off”, that is not connected to a cell j and thus not applying any force. The probabilities that P_{i} are switched “on” or “off” during the time step Δt are given by and respectively. At every time step, for each protrusion P_{i}, it is stochastically determined whether P_{i} should be updated given the probabilities (if P_{i} is currently off) or (if P_{i} is currently on). If that is the case and the current state is “on”, P_{i} is switched off. Alternatively, if the current state is off, a random cell x_{j} is chosen such that it is found at a distance from x_{i} smaller than 2r_{0}, and P_{i} is then connected to x_{j} at the other end.

Fusion and parallel plate compression simulations.
For the simulation of fusion events, two separated spherical aggregates are created by randomly generating 3D points within a sphere. Next, we let the system evolve for a short transient of time to make sure the two aggregates have reached its equilibrium configuration prior to the start of the simulation. To start the fusion process we move the aggregates closer so that they contact each other. Finally, to simulate the process of parallel plate compression we defined the position of the upper/lower plate z_{k}, k = 1, 2 as the position of the uppermost/lowermost cell over time along the z-axis. A cell i in the aggregate will experience a force F^{c}_{ik} from plate k defined as

where z_{ki} = z_{k} − z_{i} is the distance between plate k and a cell i along the z-axis, is the number of cells that fulfil the condition 2r_{0} − |z_{ki}| > 0 (i.e. interact with plate k) at time t, and F is the total external force applied to each plate. The extended dynamics of each cell i reads

The initial setup consists of a single spherical aggregate of cells and two plates positioned on opposite sides of the aggregate along the z-axis. At the start of the simulation and during a certain time period, an external force of magnitude F is applied in each plate in order to compress the aggregate. After that period, the plates are removed so that relaxation can take place (see Movies S5 and S6, ESI†).

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## Footnote |

† Electronic supplementary information (ESI) available. See https://doi.org/10.1039/d2sm00063f |

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