Fluidization and anomalous density fluctuations in 2D Voronoi cell tissues with pulsating activity

Significance

The nonmotility activities, e.g., cell pulsation, are generic features of cell tissues. However, how this activities affect the structure, dynamic, and density fluctuations of cell tissues remains poorly understood. In this work, based on a hydrodynamic theory and simulations, we show that cell pulsation can generate two opposite anomalous fluctuating states, i.e., the hyperuniform fluid state and giant density fluctuation (anti-hyperuniform) state, depending on the synchronization degree and period of pulsation. We also find that the pulsation synchronization can induce a Berezinskii–Kosterlitz–Thouless-type liquid–solid transition, in which topological defects in the pulsating phase space dominate the dynamic and density fluctuations of cell tissues. These findings deepen our understanding of recent experimental observations in Madin-Darby canine kidney monolayers.

Abstract

Cells not only can be motile by crawling but are also capable of nonmotility active motions like periodic contraction or pulsation. In this work, based on a Voronoi cell model, we show how this nonmotility activity affects the structure, dynamic, and density fluctuations of cellular monolayers. Our model shows that random cell pulsation fluidizes solid epithelial tissues into a hyperuniform fluid state, while pulsation synchronization inhibits the fluidity and causes a reverse solidification. Our results indicate this solidification is a Berezinskii–Kosterlitz–Thouless-type transition, characterized by strong density/dynamic heterogeneity arising from the annihilation of topological defects in the pulsating phase space. The magnitude and length scale of density heterogeneity diverge with the pulsating period, resulting in an opposite giant density fluctuation or anti-hyperuniformity. We propose a fluctuating hydrodynamic theory that can unify the two opposite anomalous fluctuation phenomena. Our findings can help to understand recent experimental observations in Madin-Darby canine kidney monolayers.

Collective motion of cells in epithelial tissues underlies many vital life processes, like embryonic development, wound healing, and cancer invasion (1–3). These processes are usually associated with the fluidization of epithelial tissues (4–7) or flocking of cells (8, 9), etc. (10–12). Previous studies focused on the role of crawling-induced motility of cells in these processes (6, 8–10, 13–16). However, nonmotility activities, e.g., active cell deformation or periodic pulsation (17–22), are also generic features of cell tissues, which generate active reciprocal force or stress between cells. For example, pulsating contraction induced by actin and myosin dynamics (23–28) are closely related to morphogenesis, pattern formations, and contraction waves in epithelial tissues (29–43). Meanwhile, pulsating activity can be caused by the exchange of fluid between cells (44, 45) or active nuclear movements (46). During these active pulsating processes, synchronization has been observed (36, 44). How pulsating activities and their synchronization affect the fluidity of epithelial tissues remains poorly understood. Moreover, epithelial tissues show giant density fluctuations driven by collective cell migration or cell pulsation (44, 47), while in other cellular systems, suppressed density fluctuations or hyperuniform states were reported (48–52). Thus, it is also intriguing to reveal the connection between pulsating activities and anomalous density fluctuations in epithelial tissues.

In this work, we theoretically study the effects of pulsating activity on the structure, dynamic, and density fluctuations of epithelial monolayers. Based on a Voronoi cell model, we find that random cell pulsation fluidizes epithelial tissues, resulting in a hyperuniform fluid state of class I hyperuniformity (53–56). When cell pulsations are synchronized, the fluid state is reversely solidified. Our analysis indicates this synchronization-induced solidification is a Berezinskii–Kosterlitz–Thouless (BKT) type transition, where density and dynamic heterogeneity emerge at the transition point due to the birth of topological defects in the pulsating phase space. Moreover, we find both the magnitude and length scale of the density heterogeneity diverge with the pulsating period, which results in an opposite giant density fluctuation (57, 58) or anti-hyperuniformity (54, 59). We generalize the underlying mechanism of the apparently opposite fluctuation phenomena into a fluctuating hydrodynamic theory. We also demonstrate that the above phenomena are insensitive to the form of pulsation, e.g., cell volume or junction tension pulsation. Our work provides a minimal model to understand recently observed giant density fluctuations and associated topological defects in Madin-Darby canine kidney monolayer (34, 36, 44).

Results

Model.

We consider a 2D confluent cell monolayer composed of N pulsating cells whose area and shape can change with time (Fig. 1A). Cells are modeled as polygons determined by the Voronoi tessellation of the space (6). We assume the pulsation is strong enough that the effect of thermal noise can be neglected in the overdamped cell dynamics. In our model, each cell i has a normalized preferred area $a_0^i$ which does stochastic oscillation, i.e.,

$$a_0^i=1+\alpha sin{\phi }_i(t),$$ [1]

Fig. 1.

(A) Schematics of a cell configuration (denoted in red) over a cell pulsating cycle $T_0$. (B) ${\mathrm{MSD}}_c$ under fixed $p_0=3.7$ and different pulsating strength $\alpha =[0.1,0.2,0.3,0.4,0.7]$ (Bottom to Top). (C) Phase diagram of diffusion coefficient $D_{\mathrm{c}}$. The white dashed line represents $D_{\mathrm{c}}^{\ast }={10}^{-4}{\sigma }^2/\tau$. (D and E) Typical solid ($p_0=3.7,\alpha =0.2$) and fluid ($p_0=3.7,\alpha =0.6$) configurations. Blue arrows show the cell displacements after one period. All data are for random pulsation systems ($J=0$).

where α is the pulsating strength and ${\phi }_i(t)$ is the pulsating phase. The normalized preferred perimeter is $p_0^i=p_0$ with $p_0$ the so-called target shape factor (5, 6) characterizing the time-averaged preferred shape of cells. The preferred area and perimeter of cells can not be all satisfied in a confluent cell tissue, which gives rise to an effective energy that determines the dynamics of cells (see Materials and Methods for details). Moreover, to account for collective cell pulsation, we assume ${\phi }_i(t)$ obeys the stochastic Kuramoto-like dynamics (60–63), i.e.,

$$\frac{d{\phi }_i}{\mathit{dt}}={\omega }_0+J\sum _{j\in C_i}sin({\phi }_j-{\phi }_i)+\sqrt{2T_{\phi }}{\eta }_i(t),$$ [2]

where frequency ${\omega }_0$ and the corresponding period $T_0=2\pi /{\omega }_0$ are supposed to be associated with certain intrinsic cell cycles (36, 44). The second term represents the coupling between adjacent cells with J the coupling strength. $T_{\phi }$ is the noise strength in the phase space and ${\eta }_i(t)$ is the Gaussian white noise. This simplified model is inspired by more sophisticated biological models while retaining the essential physics. It should be noted that Eq. 2 is decoupled from the cellular interaction energy (Eq. 10 in Materials and Methods). Adding such a coupling essentially enables the mechanochemical feedback which can give rise to more complicated cellular dynamics (18, 22, 36, 64). In this article, we define the unit length and unit time as σ and τ, respectively. Detailed simulation information can be found in Materials and Methods.

Cell Pulsation–Induced Fluidization.

We first study the case of zero pulsation coupling ($J=0$). To characterize the dynamical states of cell tissues, we calculate the cage relative mean square displacement of cells, ${\mathrm{MSD}}_c(t)$, which excludes the influence of the long-wavelength fluctuations in 2D (Materials and Methods). As shown in Fig. 1B, the plateau of ${\mathrm{MSD}}_c$ gradually disappears and exhibits diffusion scaling with increasing α. This indicates a pulsation-induced fluidization analogous to the one induced by motility (6). Similar behaviors have been reported in the actively deforming spherical particle system (17). Microscopically, the fluidization arises from unbalanced intercellular forces induced by random pulsation, which results in frequent T1 transitions (Movie S1). In Fig. 1C, we plot the diffusion coefficient $D_{\mathrm{c}}$ of systems in dimensions of α and $p_0$, where the solid and liquid phases can be roughly separated by a threshold $D_{\mathrm{c}}^{\ast }={10}^{-4}{\sigma }^2/\tau$ (6). We also confirm that the solid phase in our model has the crystal order (SI Appendix, Fig. S1B and Movie S2). It would be interesting to further study the nature of this transition and the glass dynamics in detail (65, 66). From Fig. 1C, one can see that systems with large $p_0$ are more easily fluidized as long as $\alpha \gtrsim 0.1$. This accords with the energy barrier scenario of T1 transition (5). Nevertheless, by accurately calculating the average shape factor $\overline{q}=\sum _i(p_i/\sqrt{a_i})/N$, we find the line of zero-energy-barrier, i.e., ${\overline{q}}^{\ast }=3.813$ (5, 6), does not follow the trend of line $D_{\mathrm{c}}^{\ast }$, especially for small α. This results in a phase regime with $p_0>3.8$ and $\alpha <0.1$, in which cells are highly deformed (large $\overline{q}$) but their movements are arrested (small $D_c$). Similar states have also been observed experimentally in MDCK monolayer with low fluctuating cell junction tension (35).

Synchronization-Induced Solidification.

We further consider the intercellular coupling during pulsation. We find that increasing coupling strength J leads to a synchronization transition at $J^{\ast }\simeq 0.06{\tau }^{-1}$, above which cells pulsation are synchronized. The synchronization degree can be characterized by the phase polarity $M=⟨|\sum _i(cos{\phi }_i,sin{\phi }_i)|⟩/N$ (Fig. 2A), which ranges from [0, 1]. We find that this synchronization transition is insensitive to the choice of α and $p_0$ (SI Appendix, Fig. S2 A and C). To explore how the synchronization transition in the pulsating phase space affects the cell dynamics in real space, we further plot $D_{\mathrm{c}}$ in dimensions of J and $p_0$ at fixed $\alpha =0.4$ in Fig. 2B. We find that for the liquid phase, $D_{\mathrm{c}}$ drops quickly as J increases above $J^{\ast }$, suggesting a synchronization-induced solidification. Compared with the unsynchronized states, the cell sizes in synchronized states are more uniformly distributed (SI Appendix, Fig. S2D). Thus, cells experience a more balanced intercellular force during collective pulsation. This suppresses the T1 transition and leads to the solidification.

Fig. 2.

(A) Phase polarization M (red line) and topological defect density ${\rho }_t$ (blue line) as a function of J at fixed $p_0=4.0$, with transition point $J^{\ast }\simeq 0.06{\tau }^{-1}$. (B) Phase diagram of $D_c$ at $\alpha =0.4$. The white dashed line represents states with $M=0.5$. (C) Cell area distribution for the system at $J=0.05{\tau }^{-1}$ and $p_0=4.0$. (D) The pulsating phase field for the same configuration in (C) with the color arrows indicates the phase. The solid circle and open circle represent the +1 and −1 topological defects, respectively. (E) Instantaneous velocity field for the configuration in (C). (F–H) The same figures as that in (C–E) for a pair of topological defects at $J=0.01{\tau }^{-1}$, $p_0=4.0$, and $T_{\phi }=0$. For all simulations, $N=4,096$.

Interestingly, near the synchronization transition point, we find the coexistence of high-density contracted regions and low-density expanded regions that periodically switch their states (Fig. 2C and Movie S3). This density heterogeneity is strongest at $J^{\ast }$, as evidenced by the height of the first peak in structure factor $S(q)$ (Fig. 3C). In Fig. 2E, we also show that cell motility is negligible like a solid in the expanded regions, while much higher akin to a liquid in the contracted regions. We find that this dynamic heterogeneity is a result of heterogeneous distribution of local shape factor ${\overline{q}}_i(t)=(p_i(t)/\sqrt{a_i(t)})$ which is large (small) in the contracted (expanded) regions. A larger (smaller) shape factor corresponds to a more anisotropic (isotropic) cell shape and stronger (weaker) imbalanced intercellular forces which favor the liquid (solid) phase (5, 6). We note that similar density heterogeneities and local synchronized pulsation patterns were observed in MDCK monolayers (33, 34, 44).

Fig. 3.

(A–D) Simulation data (solid symbols) and theoretical predictions (dashed lines) of $S(q)$ for systems with (A) different $T_0$ at fixed $(p_0,\alpha )=(4.0,0.4)$ and (B) different α at fixed $(p_0,T_0)=(4.0,20\tau )$, under random pulsation ($J=0$); (C) different J at fixed $(p_0,\alpha ,T_0)=(4.0,0.4,20\tau )$; (D) different $T_0$ at fixed $(J,p_0,\alpha )=(0.4{\tau }^{-1},4.0,0.4)$. In (C and D), a correction $\chi =0.25$ is used; (E and F) Simulation data of $S_{\text{max}}$ (red solid symbols) and $q^{\ast }$ (blue hollow symbols) as well as the corresponding theoretical predictions (dashed lines) for systems with (E) different J at fixed $(\alpha ,T_0)=(0.4,20\tau )$; (F) different $T_0$ at fixed $(J,\alpha )=(0.4{\tau }^{-1},0.4)$. Note that no χ correction is used in the theoretical predictions in (E and F). Here, $N=16,384$ for all simulations. $\gamma =1.1$ and simulation data of M are used for all theoretical predictions.

It is known that the (de)synchronization transition described by the lattice Kuramoto model with monodispersed frequency belongs to BKT universality, accompanied by the creation and annihilation of (±1) topological defects (63). We confirm this scene by calculating the topological defect density ${\rho }_t=Q/N$ in the pulsating phase space (Fig. 2A). Furthermore, we plot the pulsating phase field of the cell monolayer near the critical point in Fig. 2D. We identify $\pm 1$ topological defects and find they are mostly located on the boundaries between contracted and expanded regions. This kind of topological defect has recently been identified in MDCK monolayer in experiments (36). To further disclose the relationship between the topological defects and density/dynamic heterogeneities, we create a single pair of $\pm 1$ topological defects in the pulsating phase space and observe its evolution (Fig. 2F–H). We find that the topological defects create a synchronized cell droplet in between, which contracts (fluidizes) or expands (solidifies) alternatively. It also creates a local pulsating wave around the defect pair (Movie S4). Thus, the synchronization-induced solidification of the monolayers is essentially a BKT-type transition, resulting from the annihilation of topological defects in the pulsating phase space.

Anomalous Density Fluctuations.

Next, we focus on the density fluctuations of the pulsation system, which can be characterized by the static structure factor $S(q)$. As shown in Fig. 4, $S(q)$ for random pulsating tissue ($J=0$) in liquid state ($\alpha =0.4$) shows a scaling $S(q\to 0)\sim q^2$, which is a hallmark of class I hyperuniform state with vanishing density fluctuations at infinite wavelength (53, 54). Similar hyperuniform fluid states were also found in circle swimmers and active spinners systems (55, 56, 67). In order to study the effect of pulsating period $T_0$ on hyperuniformity, we plot the structure factor $S(q)$ for random pulsation systems under different $T_0$ and fixed $(p_0,\alpha )=(4.0,0.4)$ in Fig. 3A. With decreasing $T_0$, we find that the $q^2$ scaling remains unchanged, while the degree of hyperuniformity is enhanced. Decreasing α has a similar effect (Fig. 3B), but the relaxation dynamic would also be slower. Thus, hyperuniformity is the strongest when cells undergo fast and weak pulsation in the liquid state.

Fig. 4.

Finite size effect analysis of static structure factors for the system with random pulsation ($\alpha =0.4$, solid symbols) and without pulsation ($\alpha =0$, open symbols). Inset: The effective hyperuniformity index $H=S(q\to 0)/S(q_{\text{peak}})$ as a function of N for the two cases. Other parameters are $(p_0,T_0)=(3.8,20\tau )$.

In addition to the pulsating period $T_0$ and strength α, the synchronization of pulsation also affects the density fluctuations of the system. With increasing J from zero to $J^{\ast }$ for system under $(p_0,\alpha )=(4.0,0.4)$, we find that a peak in $S(q)$ gradually develops around $q^{\ast }\sim 0.3 {\sigma }^{-1}$ without modifying the $q^2$ scaling at small q regime (Fig. 3C). This indicates the coexistence of local large density fluctuations with global hyperuniformity, a scenario similar to circle swimmer systems (55). Interestingly, with further increasing J, the height of the peak denoted as $S_{\text{max}}$ gradually decreases. This is best shown in Fig. 3E, where we plot $S_{\text{max}}$ as a function of J for two different $p_0$. We find $S_{\text{max}}$ exhibits diverging behavior when $J\to J^{\ast }$, which is more pronounced for large $p_0$. In Fig. 3D, we further show $S(q)$ for various $T_0$ in synchronized states with $J=0.4{\tau }^{-1}$. With increasing $T_0$, $S_{\text{max}}$ increases, and the location of the peak $q^{\ast }$ shifts to the small q. This indicates that both the magnitude and length scale of the density heterogeneity diverge when $T_0\to \infty$, resulting in giant density fluctuations or anti-hyperuniformity following the reverse $q^{-2}$ scaling. This mechanism is quite robust, independent of whether the system is in the solid or liquid phase (SI Appendix, Fig. S3). It should be mentioned that giant density fluctuation was reported in MDCK monolayer experimentally (44). Our model is the minimal model that can qualitatively reproduce this phenomenon (Movie S5).

It should be emphasized that the hyperuniform fluid state induced by cell pulsation is distinct from previously reported static effective hyperuniform state (corresponding to $\alpha =0$ in our model) (50, 51). As a comparison, in Fig. 4, we plot the structure factor for systems with random pulsation ($\alpha =0.4$) and without pulsation ($\alpha =0$) under different system sizes and fixed $(p_0,T_0)=(3.8,20\tau )$. One can see that while the pulsating cellular system shows system-size independent $q^2$ hyperuniform scaling, the nonpulsating cellular system ($\alpha =0$) only shows a certain degree of uniformity. This is further confirmed by the Inset of Fig. 4, where we compare the effective hyperuniformity index $H=S(q\to 0)/S(q_{\text{peak}})$ (51) of two systems. One can see that H of pulsating cell systems vanishes as the system size increases ($H(N)\sim N^{-1}$), which is a characteristic of a strict hyperuniformity, while H remains a constant in the nonpulsating case. In fact, it was suggested that the nonpulsating system can only show strict hyperuniformity in the case of zero perimeter modulus ($K_P=0$) due to the equal-area packing of cells (68, 69), which is difficult to achieve in real cell tissue. Moreover, in SI Appendix, Fig. S4, we demonstrate that this hyperuniform fluid state is dynamically robust to external perturbation. We also give the response of hyperuniform fluid state to thermal noise in SI Appendix, Fig. S5 which accords with ref. 55. In the next section, we will reveal the hydrodynamic mechanism behind the hyperuniform fluid state, as well as the giant density fluctuation state.

Fluctuating Hydrodynamic Theory of Pulsating Cell Tissue.

To understand the above anomalous density fluctuations, we construct a hydrodynamic theory based on cell density field $\rho (\mathbf{r},t)$ and pulsating phase field $\phi (\mathbf{r},t)$(SI Appendix),

$${\partial }_t\rho =\frac{\mu }{{\rho }_0}\mathrm{\nabla }·(\rho \mathrm{\nabla }P)+A_{\alpha }{\mathrm{\nabla }}^2\eta (\mathbf{r},t),$$ [3]

$${\partial }_t\phi ={\omega }_0+\sqrt{3}\frac{J}{{\rho }_0}{\mathrm{\nabla }}^2\phi +\sqrt{2T_{\phi }}\xi (\mathbf{r},t),$$ [4]

where,

$$\begin{array}{cc}\hfill P(\rho ,\phi ,M)& =-2K_A\left[\frac{1}{\rho }-\frac{1}{{\rho }_0}\left(1+\alpha Msin\phi \right)\right]\hfill \\ \hfill & +K_P{p}_0^2\left(\frac{\sqrt{\rho }}{\sqrt{{\rho }_0}}-1\right)\hfill \end{array}$$ [5]

is the local pressure field. For a homogeneous state $\rho ={\rho }_0$, Eq. 5 can be simplified into $P_0=\frac{2K_A}{{\rho }_0}\alpha Msin\phi$, which reflects that pulsation generates periodic active stress in the cell tissue that becomes zero in the random pulsating case ($M=0$) under coarse-graining. The second term in Eq. 3 is the center-of-mass conserved noise terms (55, 56, 70) caused by the active reciprocal interaction, i.e., cell pulsation. The noise strength is $A_{\alpha }=\gamma \alpha (1-M)\sqrt{T_0}$ for small $T_0$. Eq. 4 is the coarse-graining of Eq. 2. η and ξ are two uncorrelated Gaussian white noise. It should be mentioned that the mechanochemical feedback mechanism can also be considered in the framework of hydrodynamic theory by adding a term like $-{\partial }_{{\phi }_i}ϵ$ in Eq. 4, which was shown to lead to dynamic instability patterns, like propagating waves (18, 64). By making a weak perturbation around the synchronous homogeneous state, i.e., $\rho (\mathbf{r},t)={\rho }_0+\delta \rho (\mathbf{r},t)$, $\phi (\mathbf{r},t)={\omega }_{0}t+\delta \phi (\mathbf{r},t)$, we obtain equations in the Fourier space with the linear approximation,

$$\begin{array}{cc}\hfill i\omega \delta {\rho }_{\omega ,\mathbf{q}}& =-D^{\prime }{q}^2\delta {\rho }_{\omega ,\mathbf{q}}-A_{\alpha }{q}^2{\eta }_{\omega ,\mathbf{q}}\hfill \\ \hfill & -{\alpha }^{\prime }{q}^2(\delta {\phi }_{\omega +{\omega }_0,\mathbf{q}}+\delta {\phi }_{\omega -{\omega }_0,\mathbf{q}}),\hfill \end{array}$$ [6]

$$i\omega \delta {\phi }_{\omega ,\mathbf{q}}=-J^{\prime }{q}^2\delta {\phi }_{\omega ,\mathbf{q}}+\sqrt{2T_{\phi }}{\xi }_{\omega ,\mathbf{q}},$$ [7]

where $[\delta {\rho }_{\omega ,\mathbf{q}},\delta {\phi }_{\omega ,\mathbf{q}}]=\int dt{e}^{-i\omega t}\int d\mathbf{r}{e}^{-i\mathbf{q}·\mathbf{r}}[\delta \rho ,\delta \phi ]$. ${\alpha }^{\prime }=\frac{\alpha M\mu K_A}{{\rho }_0}$ is the effective pulsation strength. $J^{\prime }=\frac{\sqrt{3}}{{\rho }_0}J$ and $D^{\prime }=\frac{\mu }{{\rho }_0}\left(\frac{2K_A}{{\rho }_0}+\frac{K_P{p}_0^2}{2}\right)$ are the effective coupling strength and diffusion coefficient respectively. From Eqs. 6 and 7, we can obtain

$$S(q)=2T_{\phi }\frac{{\alpha }^{\prime 2}(J^{\prime }+D^{\prime })}{J^{\prime }{D}^{\prime }}\frac{q^2}{{(J^{\prime }+D^{\prime })}^2{q}^4+{\omega }_0^2}+\frac{A_{\alpha }^2{q}^2}{2D^{\prime }}.$$ [8]

In the case of random pulsation ($M=0$ and ${\alpha }^{\prime }=0$), $S(q)$ exhibits a $q^2$ hyperuniform scaling (54), as a result of the emergent long-range hydrodynamic correlation (56). In Fig. 3 A and B, we plot the theoretical prediction as the dashed line, which roughly matches the simulation data under different α and $T_0$ by setting the adjustable parameter $\gamma =1.1$ in $A_{\alpha }$. In the case of synchronized state ($M\simeq 1$ and $A_{\alpha }\simeq 0$), the theory predicts the emergence of the first peak in $S(q)$ at finite $T_0$ and giant density fluctuations with $q^{-2}$ scaling as $T_0\to \infty$. In Fig. 3 E and F, we plot $S_{\text{max}}$ and $q^{\ast }$ of these first peaks as a function of J and $T_0$, respectively. One can see that theoretical lines match the simulation data without additional fitting parameters for the synchronized solid phase ($p_0=3.7$). Especially, the theory predicts the right diffusion scaling relationship $q^{\ast -2}\simeq (J^{\prime }+D^{\prime })T_0$. For the large deformation case ($p_0=4.0$), our current theory overestimates the local pressure contributed by the junction tension of cells. Therefore, a correction $\chi (p_0)<1$ is needed to match the simulation data (Fig. 3 C and D and SI Appendix). From Eq. 8, one can conclude that the center-of-mass conserved noise arising from active reciprocal interaction and periodically driving are two important ingredients to form hyperuniform fluid in the pulsation cellular systems (55, 56, 71–74).

Generalization to Other Pulsation Form.

In the above, we focus on cell pulsation driven by volume/area oscillation. Biologically, cell pulsation might also come from fluctuating junction tension (24, 26, 27, 32, 35, 37). Thus, we extend our studies to the case where $p_0^i$ does stochastic pulsation as

$$p_0^i=p_0[1+\beta cos{\phi }_i(t)]$$ [9]

while $a_0^i=1$. Here, β is the pulsating strength. This tension pulsation system exhibits a similar fluidic hyperuniformity under random pulsation (Fig. 5A and SI Appendix, Fig. S6 A and B), as well as synchronization-induced dynamical slowdown and giant density fluctuations at large pulsating period (Fig. 5B and SI Appendix, Fig. S6 C and D). In addition, topological defects in pulsating phase space still underlie the structure and dynamic heterogeneity of cell tissue (SI Appendix, Fig. S7). Unlike area pulsation, perfectly synchronized tension pulsation is similar to global cyclic shear, which destabilizes the ordered solid state (Movie S6). Finally, the hydrodynamic theory can also be extended to this pulsating scenario and qualitatively explain the two anomalous density fluctuation phenomena (see SI Appendix for discussion). All these results demonstrate the robustness of the pulsating activity in determining fluidization and anomalous density fluctuations in cell tissues.

Fig. 5.

Simulation data of $S(q)$ for tension pulsating systems with (A) different pulsation strengths β at fixed $(p_0,T_0)=(4.0,20\tau )$ for random pulsation case ($J=0$); (B) different $T_0$ at fixed $(J,p_0,\beta )=(0.4{\tau }^{-1},3.7,0.12)$. Here, $N=16,384$ for all simulations.

Discussion and Conclusion

In this work, we propose an epithelial tissue model with pulsating activity. By using this model, we show that random cell pulsation generating reciprocal active forces, melts solid epithelial tissues into a hyperuniform fluid state, while pulsating synchronization induces a reverse solidification. We demonstrate that the pulsation-induced hyperuniform fluid is a strict hyperuniform state that can recover from external perturbation. Our results also indicate that the reverse solidification is a BKT-type transition and the birth of topological defects in the pulsating phase space induces strong dynamical and structural heterogeneities at the transition point. We further construct a fluctuation hydrodynamic theory which can describe two opposite anomalous density fluctuations in the same framework. Our work deepens our understanding of recently observed giant density fluctuation and topological defects in the MDCK monolayer (33, 34, 36, 44). Although we have not yet found a biological tissue satisfying the condition for fluidic hyperuniformity, we expect that genetic engineering techniques may help to create synthetic cell tissue to realize this goal (75). Moreover, the macroscopic pulsating robotic tissue (76) can also be used to test our prediction.

In addition, it would be straightforward to incorporate the mechanochemical feedback of cells in our model, which is expected to change the fluidization of tissue and corresponding BKT scenario and may induce other interesting fluctuation phenomena (18, 36, 64, 77). Moreover, it is also tempting to study pulsating cellular systems with nematic order and the relationship between nematic topological defects and tissue flow (78–81). Such exploration would help us to understand how squeezing and contractile forces sculpt cell tissues (82–84) and also guide the design of tissue-like robotic swarms (76).

Materials and Methods

Simulation Details.

Simulations are performed within a square region $L^2$ with periodic boundaries. Due to the confluent nature of epithelial tissue, we have $L=\sqrt{N{A}_0}$ with unit area $A_0$. The center of cell i obeys the overdamped dynamics, i.e. $\dot{{\mathbf{r}}_{\mathbf{i}}}=\mu {\mathbf{F}}_i$ with μ the mobility coefficient and ${\mathbf{F}}_i=-{\mathrm{\nabla }}_{i}E$ the intercellular force acting on cell i, where E is the elastic energy of cell tissue. In our simulation, the unit length, unit energy, and unit time are defined as $\sigma =\sqrt{A_0}$, ${ϵ}_0=K_A{A}_0^2$ and $\tau ={\sigma }^2/\mu {ϵ}_0$, respectively. Then, the dimensionless energy of the Voronoi model can be written as (6, 85–87).

$$\begin{array}{cc}\hfill ϵ& =\frac{E}{{ϵ}_0}\hfill \\ \hfill & =\frac{1}{K_A{A}_0^2}{\displaystyle \sum _i^N}[K_A{(A_i-A_0^i)}^2+K_P{(P_i-P_0^i)}^2]\hfill \\ \hfill & ={\displaystyle \sum _i^N}[{(\frac{A_i}{A_0}-\frac{A_0^i}{A_0})}^2+\frac{K_P}{K_A{A}_0}{(\frac{P_i}{\sqrt{A_0}}-\frac{P_0^i}{\sqrt{A_0}})}^2]\hfill \\ \hfill & ={\displaystyle \sum _i^N}[{(a_i-a_0^i)}^2+\xi {(p_i-p_0^i)}^2].\hfill \end{array}$$ [10]

Here, $K_A$ and $K_P$ are the area and perimeter modulus of cells. $A_i$ and $P_i$ are the area and perimeter of cell i, while the corresponding $A_0^i$ and $P_0^i$ are their preferred values. In this energy function, the first term on the right describes the area elasticity of the cell and the second originates from the competition between cell contractility and adhesion tension (3, 6). $\xi =K_P/(K_A{A}_0)$ is the dimensionless perimeter modulus. For area pulsation systems, we have $a_0^i=1+\alpha sin{\phi }_i$ and $p_0^i=p_0$. For tension pulsation systems, we have $a_0^i=1$ and $p_0^i=p_0(1+\beta sin{\phi }_i)$.

Our simulation starts with a random initial configuration unless otherwise stated. We use Euler’s method to integrate the equation of phase Eq. 2 and the equation of motion with a time step $\mathrm{\Delta }t=0.02\tau$. We fix $\xi =1$, $T_{\phi }=0.1{\tau }^{-1}$ and $T_0=200\tau$ in simulation unless otherwise stated. We simulate at least $10,000\tau$ to ensure that the system reaches a dynamically stable state before sampling. The simulation is performed with the number of cells N ranging from $400$ to $262,144$ depending on the measured quantities. The Voronoi tessellation is based on the C++ library CGAL (88).

Cage Relative Mean Square Displacement.

The dynamic state of cellular monolayers can be accurately characterized by the cage relative mean square displacement of cells, ${\mathrm{MSD}}_c(t)=\frac{1}{N}〈\sum _i|{\mathbf{u}}_i(t)-{\mathbf{u}}_i^{\mathit{cage}}(t)|〉$, which excludes the disturbance of long-wavelength fluctuations in 2D. Here, ${\mathbf{u}}_i(t)={\mathbf{r}}_i(t)-{\mathbf{r}}_i(0)$ is the displacement of the cell i over the time period t, and ${\mathbf{u}}_i^{\mathit{cage}}(t)=(1/n_i)\sum _{j\in C_i}{\mathbf{u}}_j(t)$ is the displacement of the ‘cage’ which is composed of $n_i$ adjacent cells. 〈〉 represents the time average. The diffusion coefficient can be calculated by $D_c= \underset{t\to \infty }{lim}[{\mathrm{MSD}}_c(t)/(4t)]$.

Static Structure Factor.

The static structure factor is defined as $S(\mathbf{q})=\frac{1}{N}〈{\left|{\displaystyle \sum _j^N}{e}^{i\mathbf{q}·{\mathbf{r}}_j}\right|}^2〉$. Here, $\mathbf{q}=[q_x,q_y]=[i\frac{2\pi }{L},j\frac{2\pi }{L}](i,j=1,2,3,...)$ and L is the box size. Then, we project the vector $\mathbf{q}$ to scalar $q=\left|\mathbf{q}\right|$ and obtain the results of $S(q)$.

Other Information.

The methods for calculation of crystal order, identification of topological defects, and preparation of single pair topological defects, as well as details of the hydrodynamic theory, can be found in SI Appendix.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Liquid state (p₀ = 3.7 and α = 0.6) in the case of random pulsation at J = 0. Cells move away from their initial locations by random drift and frequent T1 transitions.

Movie S2.

Solid state (p₀ = 3.7 and α = 0.2) in the case of random pulsation at J = 0. Cells can not escape the cage composed of neighbors and have few T1 transitions.

Movie S3.

Coexistence of high-density contracted regions and low-density expanded regions that periodically switch their states under the parameters (J, p₀, α, T₀) = (0.05τ⁻¹, 4.0, 0.4, 200τ). Color indicates cell size as in Fig. 2C of the main text.

Movie S4.

Local pulsating wave around the defect pair under the parameters (J, p₀, α, T₀) = (0.01τ⁻¹, 4.0, 0.4, 200τ) without noise (T_φ = 0). One can observe that the pair of topological defects create a synchronized cell droplet in between, which contracts (fluidizes) or expands (solidifies) alternatively. Color indicates cell size as in Fig. 2F of the main text.

Movie S5.

Collective pulsation similar to the experimental observation (9) under the parameters (J, p₀, α, T₀) = (0.05τ⁻¹, 3.0, 1.5, 20τ).

Movie S6.

The synchronized tension pulsation state under the parameters (J, p₀, β, T₀) = (0.4τ⁻¹, 4.0, 0.12, 200τ). The synchronized tension pulsation can be understood as the global cyclic shear.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information.