text:
\section*{Introduction}
The development and maintenance of tissues requires close coordination of biochemical and mechanical signaling~\cite{lecuit2011force,heisenberg2013forces,hannezo2019mechanochemical}. There is, for instance, mounting evidence for the key role played by tissue material properties and their regulation during embryonic development~\cite{petridou2019tissue}. Tissues must be able to adjust their mechanical properties in response to internal and external stimuli. In particular, epithelial tissues, which line all cavities in the body and demarcate organs, must sustain substantial mechanical stresses while also supporting numerous biological processes such as selective diffusion and absorption/secretion~\cite{ross2006histology}. In homeostasis, the tissue must maintain its shape and resist deformation while remaining flexible. The tissue must also be able to regenerate and repair itself, often with fast turnover, e.g., in gut epithelia~\cite{krndija2019active}. Furthermore, in morphogenesis, the tissue must take up a specific shape and function~\cite{wolpert2015principles}. During metastasis, however, the shape is lost and cancer cells invade surrounding healthy tissue~\cite{weinberg2013biology}. All of these processes require that cells be able to move, often over distances much larger than the cell size. During cell migration, however, the tissue must maintain its integrity. It is, therefore, not surprising that tissues exhibit rich viscoelastic behavior~\cite{forgacs1998viscoelastic}. Unlike passive viscoelastic materials, a tissue can actively tune its rheological response, making the study of its rheology not only important for understanding biological functions but also an interesting problem from the perspective of the physics of active matter systems~\cite{marchetti2013hydrodynamics}.

Collective cell migration has been extensively studied in biology~\cite{friedl2009collective} and biophysics~\cite{alert2020physical}. In vitro studies of confluent cell monolayers~\cite{poujade2007collective,trepat2009physical,tambe2011collective,brugues2014forces,etournay2015interplay} focused on the physical aspects of force generation and transmission and showed that cell migration is an inherently collective phenomenon. Some aspects of collective cell migration are remarkably similar to slow dynamics of structural glasses~\cite{angelini2011glass,park2015unjamming,bi2015density, bi2016motility,atia2018geometric,sussman2018anomalous,czajkowski2019glassy}. This suggests that many of the observed behaviors share common underlying mechanisms and can be understood, at least at mesoscales (i.e., distances beyond several cell diameters), using physics of dense active systems~\cite{henkes2020dense}. A particularly intriguing observation is that tuning cell density~\cite{szabo2006phase,angelini2011glass,sadati2013collective}, strength of cell-cell and cell-substrate interactions~\cite{garcia2015physics}, or cell shape parameters~\cite{bi2016motility,merkel2018geometrically} can cause the collective migration to stop. In other words, the tissue undergoes fluid to solid transition. Signatures of such behavior have been reported in several in vitro~\cite{park2015unjamming,mitchel2020primary} and developmental systems~\cite{benazeraf2010random,lawton2013regulated,mongera2018fluid}. This suggests that important aspects of morphogenetic development might rely on tissue's ability to undergo phase transitions~\cite{petridou2019tissue}.

How a tissue responds to external and internal mechanical stresses will depend on its rheological (i.e., material) properties. While there have been numerous studies focusing on the rheology of a single cell~\cite{desprat2006microplates,salbreux2012actin,berthoumieux2014active}, much less is known about tissue rheology, particularly during development. In order to develop a comprehensive understanding of tissue mechanics, such insight is key. Though single cell measurements are valuable, the mechanics of a tissue can be drastically different from that of its constituent cells. The stiffness of cell monolayers, for example, is orders of magnitude higher than the stiffness of constituent cells, while the time dependent mechanical behaviors of monolayers in response to deformation vary depending on the magnitude of loading~\cite{harris2012characterizing}. Embryonic cell aggregates have been shown to behave elastically (i.e., solid-like) at short time scales but flow like fluids at long time scales, which facilitates both the robustness needed to maintain integrity and the flexibility to morph during development~\cite{forgacs1998viscoelastic}. Experiments have characterized the mechanical behaviors of tissues at various loading conditions, which led to a phenomenological description that models the relaxation properties of epithelial monolayers based on fractional calculus~\cite{bonfanti2020unified}. Notably, a recent particle-based model that includes cell division and apoptosis provided a plausible microscopic model for nonlinear rheological response~\cite{matoz2017nonlinear}. Particle-based models are, however, unable to capture geometric aspects such as cell shape. It is, therefore, necessary to investigate rheological response in geometric models.

The vertex model~(VM)~\cite{nagai2001dynamic,farhadifar2007influence,fletcher2014vertex} and more recent, closely related self-propelled Voronoi models~(SPV)~\cite{bi2016motility,barton2017active} have played an important role in modeling mechanics of epithelial tissues since they account for the shapes of individual cells and provide a link to cellular processes, such as cell-cell adhesion, cell motility, and mitosis~\cite{fletcher2014vertex}. These geometric models are also able to capture the solid to fluid transition and demonstrate rich and unusual nonlinear mechanical behavior~\cite{moshe2018geometric,popovic2020inferring}. While the mechanical properties of the VM and SPV have been extensively studied, most works to date focused on the long-time behavior, e.g., by studying the quasistatic shear modulus~\cite{bi2015density}, effective diffusion constant of cells that is related to the viscosity of tissues~\cite{bi2016motility}, and correlations between a structural property called ``softness'' and the likelihood of topological rearrangements of cells~\cite{tah2021quantifying}. The rheological properties of the VM that cover a broad range of time scales, however, have not yet been systematically explored. In this paper, we present a detailed study of the rheology of the VM by studying its response to applied oscillatory shear and bulk deformations of small amplitude, i.e., in the linear response regime. We measured the response stresses and used them to compute the storage and loss moduli in both the solid and fluid phases. We show that the dynamical response of the VM can be fitted to standard spring-dashpot viscoelastic models over seven decades in the driving frequency and that the solid-fluid transition can be tuned by applying pre-deformation to the system. Thus we argue that the VM makes a suitable basis for studies of tissue dynamics beyond the quasistatic limit.

\section*{\label{results}Results}
\noindent{\bf Vertex model dynamics.} In the VM, the state of an epithelial tissue is approximated as a polygonal tiling of the plane. The degrees of freedom are vertices, i.e., meeting points of three or more cell-cell junctions. In the simplest formulation, junctions are assumed to be straight lines. The energy of the VM is a quadratic function of cell areas and perimeters~\cite{farhadifar2007influence}, i.e.,
\begin{equation}
    E=\sum_{C}\left[\frac{K_C}{2}\left(A_C-A_{C0}\right)^2+\frac{\Gamma_C}{2}\left(P_C-P_{C0}\right)^2\right], \label{eq:vm_energy}
\end{equation}
where $K_C$ and $\Gamma_C$ are the area and perimeter elastic moduli. $A_C$ and $A_{C0}$ are the actual and preferred areas of cell $C$, respectively. Similarly, $P_C$ and $P_{C0}$ are, respectively, the actual and preferred perimeters of the same cell. In this work, we assumed $K_C$, $\Gamma_C$, $A_{C0}$, and $P_{C0}$ to be identical for all cells (i.e., $K_C\equiv K,\Gamma_C\equiv\Gamma,A_{C0}\equiv A_0,P_{C0}\equiv P_0$). Furthermore, we fixed the values of $K$ and $A_0$, and measured the energy in units of $KA_0^2$, stresses in units of $KA_0$, and lengths in units of $A_0^{1/2}$. Since the ratio between the perimeter and area elastic moduli does not qualitatively change the behavior of the VM~\cite{farhadifar2007influence,bi2015density}, we fixed that ratio to $\Gamma/(KA_0)\approx0.289$. The only variable parameter in simulations was the preferred cell perimeter $P_0$, which sets the dimensionless cell-shape parameter, defined as the ratio $p_0=P_0/\sqrt{A_0}$.

The cell-shape parameter, $p_0$, plays a central role in determining whether the system behaves as a fluid or solid~\cite{bi2015density}. Bi, \emph{et al}.~\cite{bi2015density} argued that the rigidity transition occurs at $p_0=p_c\approx3.812$ for a random polygonal tiling. The transition point is, however, at $p_c=\sqrt{8\sqrt{3}}\approx3.722$ for a regular hexagonal tiling~\cite{staple2010mechanics}. In the fluid phase, the energy barrier for neighbor exchanges vanishes and cells can flow past each other~\cite{bi2014energy}. As $p_0$ is reduced below $p_c$, the energy barrier becomes finite, neighbor exchanges cease and the system becomes solid. While the transition point for hexagonal tilings can be understood in terms of the mechanical stability and the excess perimeter~\cite{moshe2018geometric}, the mechanism that leads to a larger value for random tilings is more subtle and not fully understood ~\cite{yan2019multicellular}.

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{Fig1.pdf}
    \caption{Loss and storage shear moduli in the solid (top row) and fluid phase (bottom row). An overlay of the representative reference (grey) and sheared (yellow) configurations in (a)~the solid and (b)~the fluid phase. The magnitude of the shear is highly exaggerated for demonstration purposes. (c-d) show the representative storage ($G^\prime$) and loss ($G^{\prime\prime}$) shear moduli as functions of the shearing frequency, $\omega_0$, for different values of the cell-shape parameter, $p_0$, where we removed the dissipative contribution to $G^{\prime\prime}$ due to the velocity field resulting from the affine part of the deformation that dominates at large frequencies (see text).
    Dashed curves are the fits based on (c)~the Standard Linear Solid (SLS) model in the solid phase and (d)~the Burgers model in the fluid phase. (e-f) The collapse of the moduli curves for different values of $p_0$ for (e)~the solid phase and (f)~the fluid phase. The insets show the representation of (e)~the SLS model and (f)~the Burgers model in terms of the springs and dashpots. The majority of the data corresponds to the system of nearly square shape with $N_x=15$ cells in the horizontal direction, and we also show examples of larger systems with $N_x=37$ and $N_x=51$ cells in the horizontal direction.}
    \label{fig:simpleShear_rheology}
\end{figure*}

For definiteness, in all simulations we always started from a regular hexagonal tiling in a nearly square box subject to periodic boundary conditions, where the initial cell areas $A_C$ matched the preferred areas $A_0$ (see Methods). For the solid phase with $p_0\lesssim3.722$, a hexagonal tiling is the ground state of the energy in Eq.~(\ref{eq:vm_energy})~\cite{staple2010mechanics} and it was directly used to investigate rheological properties. Note that there was some residual hydrostatic stress (due to the mismatch of cell perimeters $P_C$ from the preferred values $P_0$), which could be eliminated by the appropriate relaxation of the simulation box. This hydrostatic stress, however, does not qualitatively affect the rheological behavior of the system (see Supplementary Information, Sec.~\ref{SI:hydrostatic_stress} for further discussion). For the fluid phase with $p_0\gtrsim3.722$, the hexagonal tiling corresponds to a saddle point of the energy in Eq.~(\ref{eq:vm_energy})~\cite{staple2010mechanics}. Thus a small random perturbation was applied to each vertex by a displacement drawn from the Gaussian distribution with zero mean and standard deviation $1.5\times10^{-4}\sqrt{A_0}$ (see Methods) and then the system was relaxed using the FIRE algorithm~\cite{bitzek2006structural} to reach a local energy minimum. Note that the energy landscape in the fluid phase has many local minima and a large number of soft modes (see Supporting Information, Sec.~\ref{SI:dynamical_matrix}). Thus we repeated simulations to investigate rheological properties for multiple configurations corresponding to different local energy minima (see Methods).

In order to probe the dynamical response of the VM, we need to specify the microscopic equations of motion for vertices. Assuming the low Reynolds number limit, which is applicable to most cellular systems due to their slow speed, inertial effects can be neglected~\cite{purcell1977life}. The equations of motion are then a force balance between friction and elastic forces due to deformations of cell shapes, i.e.,
\begin{equation}\label{Eq:overdamped_dynamics}
    \gamma\dot{\mathbf{r}}_i=\mathbf{F}_i.
\end{equation}
Here, $\mathbf{r}_i$ is the position vector of vertex $i$ in a laboratory frame of reference, $\mathbf{F}_i=-\nabla_{\mathbf{r}_i}E$ is the mechanical force on vertex $i$ due to deformation of cells surrounding it, $\gamma$ is the friction coefficient, and dot denotes the time derivative. In simulations we fixed the value of $\gamma$, which sets the unit of time as $\gamma/\left(KA_0\right)$.

We note that a precise model for dissipation in epithelial tissues is at present not known. While having a mechanism of dissipating the external energy input is central in this study, its precise microscopic details are, however, not important. We therefore assume that each vertex experiences dissipative drag proportional to its instantaneous velocity, which is a common assumption in discrete models of soft materials~\cite{frenkel2001understanding}. Furthermore, we neglected thermal fluctuations of vertices and hence omit the stochastic term in Eq.~(\ref{Eq:overdamped_dynamics}). This is a reasonable assumption since typical energy scales in tissues significantly exceed the thermal energy, $k_BT$, at room temperature $T$, where $k_B$ is the Boltzmann constant. It is important, however, to note that in epithelia there are other sources of stochasticity (e.g., fluctuations of the number of force-generating molecular motors) that are important for tissue scale behaviors~\cite{curran2017myosin}. Here, we do not consider such effects but point out that they could be directly included in the model as additional forces in Eq.~(\ref{Eq:overdamped_dynamics}).

\noindent{\bf Response to a shear deformation.} The hexagonal ground state in the solid phase and states corresponding to local energy minima in the fluid phase were then used to investigate the rheological behavior by applying an oscillatory affine shear deformation (Fig. \ref{fig:simpleShear_rheology}a,b). At each time step, we first applied the affine shear deformation to the simulation box and all vertices that was followed by internal relaxation of vertices according to Eq.~(\ref{Eq:overdamped_dynamics}) (see Methods). The affine shear deformation can be described by a deformation gradient tensor defined as $\hat{\boldsymbol{F}}=\partial\mathbf{x}/\partial\mathbf{X}_0$, where the mapping $\mathbf{x}=\mathbf{x}\left(\mathbf{X}_0,t\right)$ maps the reference configuration $\mathbf{X}_0$ to a spatial configuration $\mathbf{x}$ at time $t$. For simple shear, the deformation gradient tensor is $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}1 & \epsilon(t)\\ 0 & 1\end{smallmatrix}\big)$, where $\epsilon\left(t\right)=\epsilon_0\sin\left(\omega_0 t\right)$. Sufficiently small amplitude $\epsilon_0=10^{-7}\ll1$ was used to probe the linear response properties.

The response stress tensor, $\hat{\boldsymbol{\sigma}}_C\left(t\right)$, for each cell $C$ was computed using the formalism introduced in \cite{nestor2018relating} (see Supplementary Information, Sec.~\ref{SI:stress}). The average stress tensor $\hat{\boldsymbol{\sigma}}\left(t\right)=\sum_C w_C\hat{\boldsymbol{\sigma}}_C\left(t\right)$, with $w_C = A_C/\sum_C A_C$, was used as a measure for the response of the system. The dynamic shear modulus $G^*\left(\omega_0\right)=\tilde{\tau}(\omega_0)/\tilde{\epsilon}(\omega_0)$ was then calculated at a given frequency $\omega_0$ of applied shear strain, where $\tilde{\tau}(\omega)$ and $\tilde{\epsilon}(\omega)$ are the Fourier transforms of the response shear stress $\tau(t)=\hat{\sigma}_{xy}\left(t\right)$ and the applied strain $\epsilon(t)$, respectively (see Methods). We ensured that the simulations were sufficiently long for a steady state to be reached (see Methods and Supplementary Information, Sec.~\ref{SI:steady_state}). The real part of the dynamic shear modulus, $G^\prime=\text{Re}\left(G^*\right)$, is the storage shear modulus and the imaginary part, $G^{\prime\prime}=\text{Im}\left(G^*\right)$, is the loss shear modulus. The storage shear modulus corresponds to the in-phase response and measures the elastic (i.e., reversible) response of the system, while the loss shear modulus corresponds to the out-of-phase response and measures the irreversible dissipation~\cite{larson1999structure} (see also Supplementary Information, Sec.~\ref{SI:rheology}). Note that the stress contribution due to the friction with the substrate resulting from the part of the velocity field due to the externally imposed affine deformation was removed in the analysis. This is because at high frequencies, the stresses are dominated by the friction due to external driving that masks the internal material response that we are interested in exploring (see Fig.~\ref{fig:moduli_affineVelocity} and Supplementary Information, Sec.~\ref{SI:stress}).
For systems under an oscillatory simple shear, storage and loss shear moduli were obtained for different values of $p_0$ and different system sizes in the solid and the fluid phases with frequencies $\omega_0$ of the applied shear strain spanning over seven orders of magnitude, as shown in Fig.~\ref{fig:simpleShear_rheology}c,d. Most simulations were performed for systems with nearly square shapes with $N_x=15$ cells in the horizontal direction. We repeated several simulations for systems with $N_x=37$ and $N_x=51$, which showed that the finite size effects are negligible (Fig.~\ref{fig:simpleShear_rheology}c-f).

In the solid phase there are two different regimes (see Fig.~\ref{fig:simpleShear_rheology}c). At low frequencies, $\omega_0$, the storage shear modulus $G^{\prime}$ has a constant value, while the loss shear modulus scales as $G^{\prime\prime}\propto\omega_0$. At high frequencies the storage shear modulus $G^{\prime}$ has a higher constant value, while the loss shear modulus scales as $G^{\prime\prime}\propto\omega_0^{-1}$. Such rheological behavior is characteristic for the Standard Linear Solid (SLS) model~\cite{larson1999structure}. Storage and loss shear moduli for the SLS model are~\cite{larson1999structure}, respectively,
\begin{subequations}
\label{G_SLS}
\begin{align}
    G_\text{SLS}^\prime\left(\omega_0\right)&=\frac{E_2+\frac{\eta_1^2}{E_1^2}\omega_0^2\left(E_1+E_2\right)}{1+\frac{\eta_1^2}{E_1^2}\omega_0^2},\label{eq:Gprime}\\
     G_\text{SLS}^{\prime\prime}\left(\omega_0\right)&=\frac{\omega_0\eta_1}{1+\frac{\eta_1^2}{E_1^2}\omega_0^2}\label{eq:Gprimeprime},
     \end{align}
\end{subequations}
where we used the representation of the SLS model (Fig.~\ref{fig:simpleShear_rheology}e, inset) that consists of a spring with elastic constant $E_2$ connected in parallel with a Maxwell element, which comprises a spring with elastic constant $E_1$ and a dashpot with viscosity $\eta_1$ connected in series.
 The above expressions in Eqs.~(\ref{G_SLS}) were used to fit the storage and loss shear moduli obtained from simulations. The fitted curves, represented with dashed lines in Fig.~\ref{fig:simpleShear_rheology}c, show an excellent match with the simulation data, indicating that the SLS model is indeed appropriate to describe the shear rheology in the solid phase. This was also confirmed in Fig.~\ref{fig:simpleShear_rheology}e, where we collapsed the storage and loss shear moduli for different values of the shape parameter, $p_0$, by rescaling the moduli and frequencies with the fitted values of spring and dashpot constants.

 As the value of the $p_0$ increases, we observe that the storage shear modulus reduces at all frequencies and that the loss shear modulus reduces at high frequencies. Furthermore the crossover between the two regimes shifts towards lower frequencies (Fig.~\ref{fig:simpleShear_rheology}c). This is because the elastic constants $E_1$ and $E_2$ decrease linearly with increasing $p_0$ and they become zero exactly at the solid-fluid transition with $p_0=p_c\approx 3.722$  (Fig.~\ref{fig:SimpleShear_constant}a).
 The dashpot constant $\eta_1$ is nearly independent of $p_0$ and scales with the friction parameter $\gamma$, which is the only source of dissipation in the VM. The crossover between the two regimes for both the storage and loss shear moduli corresponds to a characteristic time scale, $\eta_1/E_1$, which diverges as $p_0$ approaches the solid-fluid transition due to the vanishing elastic constant (Fig.~\ref{fig:SimpleShear_constant}c). Note that the values of the elastic constants $E_1$ and $E_2$ can be estimated analytically. In the quasistatic limit ($\omega_0\rightarrow 0$), the external driving is sufficiently slow that the system can relax internally. In this limit, Murisic, \emph{et al.}~showed that the storage shear modulus is
\begin{equation}\label{Eq:Gquasi}
    G^{\prime}(\omega_0\rightarrow 0)=E_2=\frac{1}{2}KA_0\big(1-[\alpha(p_0,\Gamma/KA_0)]^2\big),
\end{equation}
where $\alpha(p_0,\Gamma/KA_0)$ is a scaling factor chosen such that the hydrostatic stress vanishes once the system box size is rescaled from $L$ to $\alpha L$~\cite{murisic2015discrete}. In the high frequency limit ($\omega_0\rightarrow \infty$), on the other hand, the system follows the externally imposed affine deformation and has no time for internal relaxation. Thus, by considering the energy cost for a hexagonal tiling under affine deformation, we obtained the storage shear modulus (see Supplemental Information, Sec.~\ref{SI:tuning_phase_transition})
\begin{equation}
G^{\prime}(\omega_0\rightarrow \infty)=E_1+E_2=3\sqrt{3}\Gamma\left(1-\frac{p_0}{p_c}\right).\label{Eq:Gaffine}
\end{equation}
The above Eqs.\ (\ref{Eq:Gquasi}) and (\ref{Eq:Gaffine}) were used to extract the values of elastic constants $E_1$ and $E_2$, which showed excellent agreement with the fitted values from simulations (Fig.~\ref{fig:SimpleShear_constant}a).
\begin{figure}[tbp]
    \centering
    \includegraphics[width=0.98\columnwidth]{Fig2.pdf}
    \caption{(a-b) Fitted values of spring-dashpot models for the system under simple shear. (a)~Elastic constants as a function of target cell-shape parameter, $p_0$. In the solid phase (i.e., for $p_0 < p_c\approx3.722$), fitted values of the spring constants show excellent match with the analytical predictions obtained from Eqs.~(\ref{Eq:Gquasi}) and (\ref{Eq:Gaffine}) (dashed lines). Inset shows the spring constants near the critical point. (b)~Dashpot viscosity constants as a function of the target cell-shape parameter, $p_0$. (c-d) Characteristic time scales in (c)~the solid and (d)~fluid phase obtained from the fitted values of the elastic constant and the dashpot viscosity. The normalization factor $t^*=\gamma/(K A_0)$ sets the unit of time. For the fluid phase (i.e., for $p_0 > p_c\approx3.722$), errorbars correspond to the standard deviation for simulations that were repeated for configurations that correspond to different local energy minima.}
    \label{fig:SimpleShear_constant}
\end{figure}

 In the fluid phase, the storage and loss shear moduli show a markedly different behavior (Fig.~\ref{fig:simpleShear_rheology}d). There are three different regimes with two crossover frequencies, which correspond to two characteristic time scales. At low frequencies, $\omega_0$, the storage shear modulus $G^\prime\propto \omega_0^2$ and the loss shear modulus $G^{\prime\prime}\left(\omega_0\right)\propto \omega_0$. The storage modulus approaches $0$ for $\omega_0\to0$, which indicates that the system is indeed a fluid. At high frequencies the storage shear modulus has a constant value, while the loss shear modulus scales as $G^{\prime\prime}\left(\omega_0\right)\propto \omega_0^{-1}$. To capture this behavior we used the Burgers model, which consists of two Maxwell models connected in parallel (Fig.~\ref{fig:simpleShear_rheology}f, inset), to fit the shear moduli measured in the simulations. The storage and loss shear moduli for a Burgers model are~\cite{larson1999structure}, respectively,
 \begin{subequations}
 \begin{align}
G_\text{Burg}^\prime\left(\omega_0\right)&=\frac{p_{1} q_{1} \omega_0^{2}-q_{2} \omega_0^{2}\left(1-p_{2} \omega_0^{2}\right)}{p_{1}^{2} \omega_0^{2}+\left(1-p_{2} \omega_0^{2}\right)^{2}},\label{Gp_liq}\\
G_\text{Burg}^{\prime\prime}\left(\omega_0\right)&=\frac{p_{1} q_{2} \omega_0^{3}+q_{1} \omega_0\left(1-p_{2} \omega_0^{2}\right)}{p_{1}^{2} \omega_0^{2}+\left(1-p_{2} \omega_0^{2}\right)^{2}},\label{Gpp_liq}
\end{align}
 \end{subequations}
 where $p_{1}=\eta_{1}/E_{1}+\eta_{2}/E_{2}$, $p_2=\eta_1\eta_2/(E_1E_2)$, $q_1=\eta_1+\eta_2$, $q_2=\eta_1\eta_2(E_1+E_2)/(E_1E_2)$. The dashed curves in Fig.~\ref{fig:simpleShear_rheology}d show fits of the storage and loss shear moduli for a range of values of $p_0$, which show good agreement with simulations. Unlike for the solid phase, it is not possible to collapse the data for storage and loss shear moduli onto single universal curves because the fluid phase is characterized by two independent timescales $\eta_1/E_1$ and $\eta_2/E_2$. Thus we show two different collapses for the storage and loss shear moduli in the low frequency range (Fig.~\ref{fig:simpleShear_rheology}f) and in the high frequency range (Fig.~\ref{fig:rescale_fluid_highF} in the Supplementary Information, Sec.~\ref{SI:collapse_fluid_high_freq}).

As the value of the $p_0$ decreases, we observe that both the storage and loss shear moduli reduce at intermediate and high frequencies, but they increase at low frequencies (Fig.~\ref{fig:simpleShear_rheology}d).  We also observe that the first crossover shifts towards lower frequencies, while the second crossover remains at approximately the same frequency. This is because the elastic constants $E_1$ and $E_2$ decrease linearly toward zero as $p_0$ approaches the solid-fluid transition at $p_c\approx3.722$ (Fig.~\ref{fig:SimpleShear_constant}a). The dashpot constant $\eta_2$ also decreases linearly toward zero, while the dashpot constant $\eta_1$ increases but remains finite as $p_0$ approaches the solid-fluid transition (Fig.~\ref{fig:SimpleShear_constant}b). As a consequence, one of the characteristic time scales $\eta_1/E_1$ diverges, while the second  time scale $\eta_2/E_2$ remains finite as as $p_0$ approaches the solid-fluid transition (Fig.~\ref{fig:SimpleShear_constant}d). The diverging characteristic time scale captures the macroscopic behavior of the system, while the second time scale ($\sim \gamma/KA_0$) captures the microscopic details of the VM.
Finally, we note that the values of the spring and dashpot constants are somewhat sensitive to the local energy minimum configuration used to probe the response in the fluid phase. The errorbars in Fig.~\ref{fig:SimpleShear_constant} show standard deviation for different configurations that were obtained by using the same magnitude of the initial perturbation (see Methods). In Fig.~\ref{fig:initial_kick} in the Supplementary Information, Sec.~\ref{Sup:initial_perturbation}, we show how the values of the spring and dashpot constants are affected when configurations were obtained by using different magnitudes of the initial perturbation.

\begin{figure*}[htbp]
    \centering
    \includegraphics[width=0.95\textwidth]{Fig3.pdf}
    \caption{Loss and storage bulk moduli in the solid (top row) and fluid phase (bottom row). An overlay of the representative reference (grey) and biaxially deformed (yellow) configurations in (a)~the solid and (b)~the fluid phase. The magnitude of the bulk deformation is highly exaggerated for demonstration purposes. (c-d) show the representative storage ($B^\prime$) and loss ($B^{\prime\prime}$) bulk moduli as functions of the deformation frequency, $\omega_0$, for different values of the cell-shape parameter, $p_0$, where we removed the dissipative contribution to $B^{\prime\prime}$ due to the velocity field resulting from the affine part of the deformation that dominates at large frequencies (see text). For the solid phase in (c), the loss bulk modulus $B^{\prime\prime}\equiv0$. For the fluid phase in (d), dashed curves are the fits based on the Standard Linear Solid (SLS) model. (e-f) The collapse of the moduli curves for different values of $p_0$ for (e)~the solid phase and (f)~the fluid phase. The insets show the representation of (e)~the spring model and (f)~the SLS model in terms of the springs and dashpots.}
    \label{biaxial_rheology}
\end{figure*}

\noindent{\bf Response to bulk deformations.} We further studied the bulk rheological properties of the system by applying an oscillatory biaxial deformation (Fig.~\ref{biaxial_rheology}a,b) described by the deformation gradient $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}1+\epsilon(t)&0\\0&1+\epsilon(t)\end{smallmatrix}\big)$,
where $\epsilon(t)=\epsilon_0\sin\left(\omega_0 t\right)$. We applied a sufficiently small amplitude $\epsilon_0=10^{-7}\ll1$ to probe the linear response properties characterized by the average normal stress $\sigma(t)=\frac{1}{2}\left[\hat \sigma_{xx}(t)+\hat \sigma_{yy}(t)\right]$, where we again removed the stress contribution due to the friction with the substrate resulting from the part of the velocity field due to the externally imposed affine deformation. As in the simple shear test, we then computed the dynamic bulk modulus as $B^*(\omega_0)=\tilde \sigma (\omega_0)/\tilde \epsilon(\omega_0)$ from which we obtained the storage bulk modulus $B^\prime=\text{Re}\left(B^*\right)$ and the loss bulk modulus $B^{\prime\prime}=\text{Im}\left(B^*\right)$ (see Fig.~\ref{biaxial_rheology}c,d).

In the solid phase, the storage bulk modulus is independent of the driving frequency and the loss bulk modulus is zero. Thus the response of the system can be captured by a single spring $E_\text{solid}$ (Fig.~\ref{biaxial_rheology}e, inset). This is because the hexagonal tiling is stable to biaxial deformation in the solid phase and thus there is no internal relaxation. The measured value of the storage bulk modulus matches the analytical prediction
\begin{equation}
B_\text{theory}=2KA_0+\sqrt[\leftroot{-1}\uproot{0}4]{12}\Gamma p_0
\label{eq:bulk_solid}
\end{equation}
by Staple, \emph{et al.}~\cite{staple2010mechanics},
where the hexagonal tiling is assumed to undergo affine deformation under biaxial deformation. Storage bulk moduli, normalized by $B_\text{theory}$, for different values of $p_0$ all collapse to 1 (Fig.~\ref{biaxial_rheology}e).

In the fluid phase, the bulk response behavior of the system can be described by the SLS model (Fig.~\ref{biaxial_rheology}f, inset). While it might appear counter-intuitive to model a fluid with the SLS model, this is a direct consequence of the fact that in the fluid state, the bulk modulus is finite but the shear modulus vanishes, i.e., the fluid flows in response to shear but resists bulk deformation. The fitted storage and loss bulk moduli for the SLS model [see Eq.~(\ref{G_SLS})] show an excellent match with the simulation data (Fig.~\ref{biaxial_rheology}d). This was also confirmed in Fig.~\ref{biaxial_rheology}f, where we collapsed the storage and loss bulk moduli for different values of $p_0$.
\begin{figure}[bp]
     \centering
     \includegraphics[width=0.98\columnwidth]{Fig4.pdf}
     \caption{Fitted values of spring-dashpot models for the system under bulk deformation as a function of the target cell-shape parameter, $p_0$. (a)~In the solid phase ($p_0 < p_c\approx3.722$), the bulk storage modulus $E_{\text{solid}}$ agrees with the analytical prediction $B_{\text{theory}}$ in Eq.~(\ref{eq:bulk_solid}). At the solid-fluid transition point ($p_0=p_c\approx 3.722$), it continuously changes to the high frequency limit of the bulk storage modulus, i.e., $B^{\prime}(\omega_0\rightarrow \infty)=E_1+E_2$, of the fluid phase. The low frequency limit of the bulk storage modulus is $B^{\prime}(\omega_0\rightarrow 0)=E_1$ in the fluid phase. (b)~Dashpot viscosity constant as a function of $p_0$. For the fluid phase ($p_0 > p_c\approx3.722$), errorbars correspond to the standard deviation for simulations that were repeated for configurations that correspond to different local energy minima. }
     \label{biaxial_constant}
 \end{figure}
 \begin{figure*}
    \centering
    \includegraphics[width=0.95\textwidth]{Fig5.pdf}
    \caption{Tuning the solid to fluid transition by applying uniaxial deformation. (a)~The solid-fluid transition boundary in the $a-p_0$ plane, where $a$ measures the amount of uniaxial deformation described by the deformation gradient $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}a & 0 \\ 0 & 1\end{smallmatrix}\big)$. Blue line shows the analytical prediction from Eq.~(\ref{Eq:phaseBoundary}), which  matches the stability analysis with the Hessian matrix (red dots). (b,c)~The fitted values of the (b)~spring and (c)~dashpot constants for the SLS model in the solid phase and the Burgers model in the fluid phase when the system is under uniaxial compression ($a=0.95$), no pre-deformation ($a=1$), and under uniaxial tension ($a=1.05$). }
    \label{transition_uniaxial}
\end{figure*}

The fitted values of spring elastic and dashpot viscosity constants for different values of $p_0$ are plotted in Fig.~\ref{biaxial_constant}. In the fluid phase, the storage bulk modulus in the high frequency limit ($B^{\prime}(\omega_0\rightarrow \infty)=E_1+E_2$) continuously increases from the value for the solid phase $B_{\text{theory}}$ in Eq.~(\ref{eq:bulk_solid}) as the system transitions from solid to fluid (Fig.~\ref{biaxial_constant}a). The storage bulk modulus in the quasistatic limit ($B^{\prime}(\omega_0\rightarrow 0)=E_2$) emerges at the transition point with a finite value and increases as $p_0$ increases from $p_c$ (Fig.~\ref{biaxial_constant}a). Fig.~\ref{biaxial_constant}b shows that the dashpot constant diverges as the $p_0$ decreases toward $p_c$. Thus, the characteristic time scale $\eta_1/E_1$ also diverges, but for a different reason than for the shear deformation, where the spring constant is vanishing (see Fig.~\ref{fig:SimpleShear_constant}). Finally, we note that, unlike for the response to shear, the values of the spring and dashpot constants for bulk deformation are not sensitive to the local energy minimum configuration used to probe the response in the fluid phase, which is reflected in very small errorbars in Fig.~\ref{biaxial_constant}. This is because the bulk moduli are dominated by the changes in cell areas.

{\bf Response to a shear deformation of a uniaxially pre-deformed system.} The solid-fluid transition for the regular hexagonal tiling occurs when $p_0\approx3.722$, above which the hexagonal tiling is unstable. This is consistent with the vanishing of the affine shear modulus in Eq.~(\ref{Eq:Gaffine}) at the transition point. If the regular hexagonal tiling is compressed or stretched uniaxially by a factor $a$, which is described by the deformation gradient $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}a & 0 \\ 0 & 1\end{smallmatrix}\big)$, then the high frequency limit of the storage shear modulus that is dominated by affine deformation becomes (see Supplemental Information, Sec.~\ref{SI:tuning_phase_transition}),
\begin{equation}
\begin{split}
    G^{\prime}_\text{affine}\left(a\right)=&\frac{2\sqrt{2}\Gamma}{3^{7/4}a}\left(1+\frac{1}{\left(1+3a^2\right)^\frac{3}{2}}\right)\\
    & \times \left(-3p_0+\sqrt[4]{192}\left(1+\sqrt{1+3a^2}\right)\right).
\end{split}
\end{equation}
By setting the affine shear modulus to 0, we obtained the solid-fluid transition boundary in the $a-p_0$ plane as
\begin{equation}\label{Eq:phaseBoundary}
    a\left(p_0\right) = \sqrt{\frac{\sqrt{3}p_0^2}{8}-\frac{p_0}{\sqrt[4]{12}}}.
\end{equation}
The above analytical prediction for the phase boundary (Fig.~\ref{transition_uniaxial}a, blue line) shows an excellent agreement with the stability analysis in terms of the eigenvalues of the Hessian matrix of the energy function (Fig.~\ref{transition_uniaxial}a, red dots). A given configuration is stable if all eigenvalues of the Hessian matrix are positive and the loss of mechanical stability occurs when the lowest eigenvalue becomes $0$. For a given $p_0$, the value of the lowest eigenvalue reduces with decreasing $a$, i.e., as the magnitude of compression is increased. Thus, the compression (stretching) shifts the solid-fluid transition towards the lower (higher) values of $p_0$ (see Fig.~\ref{transition_uniaxial}a).

We also probed the response to oscillatory shear applied to uniaxially compressed and stretched systems. This analysis was done on both the uniaxially deformed hexagonal tiling in the solid phase as well as a system in the fluid phase obtained by relaxing the unstable uniaxially deformed hexagonal tiling after an initial random perturbation (see Methods). The response to the shear deformation is qualitatively similar and can still be described by the SLS model in the solid phase and the Burgers model in the fluid phase. Fig.~\ref{transition_uniaxial}b,c shows fitted values of the parameters for spring-dashpot models when the system is under uniaxial compression ($a=0.95$), no pre-deformation ($a=1$, i.e., same as Fig.~\ref{fig:SimpleShear_constant}a,b), and uniaxial tension ($a=1.05$). In both the solid and fluid phases, all spring elastic constants decrease to $0$ as $p_0$ approaches the critical value predicted by Eq.~(\ref{Eq:phaseBoundary}). The dashpot viscosity remains constant in the solid phase. Once the system enters the fluid phase as $p_0$ increases, a new second dashpot constant emerges and increases from $0$, while the value of the first dashpot constant decreases. As in the simple shear case, we note that the values of the spring and dashpot constants are somewhat sensitive to the local energy minimum configuration used to probe the response in the fluid phase. The errorbars in Fig.~\ref{fig:SimpleShear_constant} show standard deviation for configurations that were obtained by using different random initial perturbation (see Methods).
Finally, we note that besides uniaxial deformation, the solid-fluid transition point can be tuned by other modes of deformation (see Fig.~\ref{fig:PhaseDiagram} and Supplementary Information, Sec.~\ref{SI:tuning_phase_transition}).

\section*{Discussion}
We have performed a detailed analysis of the rheological properties of the Vertex Model subject to small-amplitude oscillatory deformations over seven orders of magnitude in the driving frequency. Our analysis shows that the VM exhibits non-trivial viscoelastic behavior that can be tuned by a single dimensionless geometric parameter - the shape parameter, $p_0$. In order to characterize the response, we constructed constitutive rheological models that use combinations of linear springs and dashpots connected in series and in parallel. These models allowed us to match the shear response of the VM to that of the Standard Linear Solid model in the solid phase and the Burgers model in the fluid phase. In the low-frequency, i.e., quasistatic regime, our results are fully consistent with many previous studies \cite{staple2010mechanics,murisic2015discrete,bi2015density,bi2016motility}. Our work, however, provides insights into the time-dependent response of the VM over a broad range of driving frequencies, which is important if one is to develop full understanding of the rheological properties of the VM and how those inform our understanding of the rheology of epithelial tissues.

We also showed that the critical value for the solid-fluid transition can be tuned by applying pre-deformation. Interestingly, under uniaxial and biaxial (i.e., isotropic) compression the solid to fluid transition shifts to lower values of $p_0$, leading to the non-intuitive prediction that one can fluidize the system by compressing it. This is, however, not surprising since the transition is driven by a geometric parameter that is inversely proportional to the square root of the cell's native area. Compressing the system reduces its area and, hence, effectively increases $p_0$. It is, however, important to note that this is just a property of the VM and it does not necessarily imply that actual epithelial tissue would behave in the same way. Cells are able to adjust their mechanical properties in response to applied stresses and it would be too simplistic to assume that compression would directly lead to changes in the native area. In fact, experiments on human bronchial epithelial cells show that applying apical-to-basal compression, which effectively expands the tissue laterally (i.e., corresponds to stretching in our model) fluidizes the tissue~\cite{park2015unjamming}.

Furthermore, the transition from solid phase to fluid phase is accompanied by the emergence of a large number of soft modes. As we have already alluded, it has recently been argued that these soft modes lead to a nonlinear response distinct from that obtained in classical models of elasticity~\cite{moshe2018geometric}. Approximately half of the eigenmodes are zero modes (see Fig.~\ref{fig:DOS} in the Supplementary Information, Sec.~\ref{SI:dynamical_matrix}). While the analysis of soft modes in the VM is an interesting problem~\cite{yan2019multicellular}, it is beyond the scope of this work.
Other models in this class have intriguing non-trivial mechanical properties, such as the existence of topologically protected modes~\cite{kane2014topological,lubensky2015phonons,paulose2015topological,huber2016topological,rocklin2017transformable,mao2018maxwell}.

We note that while we studied the dynamical response over a wide range of frequencies, our work focused on the behavior in the linear response regime, where there are effectively no plastic events, i.e., while being allowed, T1 transformations typically did not occur during the process of probing the rheology. A full understanding of the VM rheology would also need to allow for cell rearrangements. This is, however, a very challenging problem and first steps in addressing it have only recently been made~\cite{popovic2020inferring}.

Regardless of whether cells in an epithelial tissue are arrested or able to move, the rheological response of the tissue is viscoelastic with multiple time scales~\cite{bonfanti2020unified}. This response arises as a result of the complex material properties of individual cells combined with four basic cellular behaviors: movement, shape change, division, and differentiation. The tissue not only has a non-trivial rheological response but is also able to tune it. There is growing evidence that this ability of biological systems to tune their rheology, and in particular, transition between solid-like and fluid-like behaviors, plays a key role during morphogenesis \cite{petridou2019tissue}. How such cellular processes are regulated and coordinated to form complex morphological structures is only partly understood. It is, however, clear that the process involves mechano-chemical feedback between mechanical stresses and the expression of genes that control the force-generating molecular machinery in the cell. Any models that aim to describe morphological processes, therefore, need to include coupling between biochemical processes and mechanical response. The base mechanical model, however, must be able to capture the underlying viscoelastic nature of tissues. Our work provides evidence that the vertex model, a commonly used model to study mechanics of epithelial tissues, has interesting non-trivial rheological behavior. This, combined with its ability to capture both fluid- and solid-like behavior by tuning a single geometric parameter shows it to be an excellent base model to build more complex descriptions of real tissues.

\section*{Methods}
 Most simulations were performed for systems with nearly square shapes with $N_x=15$ cells in horizontal direction (i.e., $N=240$ cells in total) subject to periodic boundary conditions. The shape of the simulation box was chosen to be as near to a square as allowed by the geometry of a hexagon and the size of the box was such that it accommodated $N$ cells of area $A_C$ that matched the preferred areas $A_0$. Simulations of larger system sizes ($N_x=37$, $51$, i.e., $N=1406$, $2652$ cells, respectively) were performed for a subset of values of $p_0$ to explore the finite size effects. No quantitative differences between the system with $N=240$ cells and larger systems were observed. Since the ratio between the perimeter and elastic moduli does not qualitatively change the behavior of the VM~\cite{bi2015density,farhadifar2007influence}, we used $\Gamma/(KA_0)\approx0.289$ for all simulations.

 We applied an oscillatory affine deformation of frequency $\omega_0$ either as simple shear described by the deformation gradient $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}1&\epsilon(t)\\0&1\end{smallmatrix}\big)$ or biaxial deformation described by $\hat{\boldsymbol{F}}=\big(\begin{smallmatrix}1+\epsilon(t)&0\\0&1+\epsilon(t)\end{smallmatrix}\big)$, where $\epsilon(t)=\epsilon_0\sin(\omega_0 t)$. In all simulations, we used a small magnitude of deformation, i.e., $\epsilon_0=10^{-7}$, so that we probed the linear response and the measured moduli were independent of the magnitude of the deformation. In every time step, the system evolved according to the overdamped dynamics in Eq.~(\ref{Eq:overdamped_dynamics}) after the affine deformation was applied. Equations of motion were integrated using the first order Euler method with the time step, $\Delta t \approx 0.00866\gamma/(KA_0)$. Measurements of the response stresses for each cell and the entire system were taken 25 times within each cycle of oscillatory deformation by using Eq.~(\ref{eq:stress_tensor}) in the Supporting Information, Sec.~\ref{SI:stress}.

To ensure that we were probing the steady state, we performed the following analysis. For example, in the case of shear deformation, the shear stress signal $\tau(t)$ was divided into blocks of length $T=3T_0$, each containing 3 cycles of the time period $T_0=2 \pi/\omega_0$ of the driving shear deformation. Within each block $n$, we performed the Fourier transform of $\tau(t)$ and obtained $\tilde{\tau}_n(\omega)$ as
\begin{equation}
  \tilde{\tau}_n(\omega)=\frac{1}{T}\int_{(n-1)T}^{nT}\tau(t)e^{i\omega t}dt,
\end{equation}
where $n$ is a positive integer. Similar Fourier transform analysis was performed for the strain, $\epsilon(t)$. The length of the simulation was chosen such that it contained a sufficient number of blocks in order for the  $\tilde{\tau}_n(\omega_0)$ to reach a steady state value  $\tilde{\tau}(\omega_0)$. The obtained steady state value of $\tilde{\tau}(\omega_0)$ was used to calculate the storage and loss shear moduli as described in the main text. Analogous procedure was applied to the normal stress, $\sigma(t)$, in the case of the bulk deformation. Please refer to the Supplementary Information, Sec.~\ref{SI:steady_state} for a representative example of the steady state analysis.

In the solid phase, we performed rheological tests on a hexagonal tiling. In the fluid phase, however, the hexagonal tiling is unstable. Instead, the hexagonal tiling was randomly perturbed and then relaxed to a nearby local stable state using the FIRE optimizer~\cite{bitzek2006structural}. The local energy minimum was determined with the relative accuracy of $10^{-12}$. A random perturbation was applied to each vertex $i$, i.e., each vertex was displaced from its original position in the hexagonal tiling by a vector $\delta \mathbf{r}_i = \delta x_i \mathbf{e}_x + \delta y_i \mathbf{e}_y$, where $x_i$ and $y_i$ were Gaussian random variables with zero mean and standard deviation $1.5\times10^{-4} \sqrt{A_0}$.
The rheology of a local stable state was then probed following the same procedure as in the solid case.

During the energy minimization and oscillatory deformations, T1 transitions were allowed but were not common. T1 transitions were implemented following the procedure introduced by Spencer, \emph{et al.}~\cite{spencer2017vertex}.

\section*{Acknowledgements}
This research was primarily supported by NSF through the Princeton
University's Materials Research Science and Engineering Center
DMR-2011750 and by the Project X Innovation Research Grant from the Princeton School of Engineering and Applied Science. RS acknowledges support by the UK BBSRC (Award
BB/N009789/1). This project was initiated during the KITP program ``Symmetry, Thermodynamics and Topology in Active Matter'' (ACTIVE20), and it is supported in part by the National Science Foundation under Grant No.\ NSF PHY-1748958. We would like to acknowledge useful discussions with Mikko Haataja.

\section*{Data availability} The data supporting the results and findings of this study is available from the corresponding authors upon reasonable request.

\section*{Code availability} Simulation and analysis codes used in this study are available from the corresponding authors upon reasonable request.

\section*{Contributions} ST, RS, and AK conceived the study and designed the project. ST and NS performed numerical simulations and analysed the data. RS developed the Vertex Model code used in numerical simulations. ST, NS, RS, and AK wrote the paper.

\section*{Competing interests}
The authors declare no competing interests.

redpajama_set_name: RedPajamaArXiv


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text:
Q: how to remove key value from map in scala Map(data -> "sumi", rel -> 2, privacy -> 0, status -> 1,name->"govind singh")

how to remove data from this map , if privacy is 0.
Map(rel -> 2, privacy -> 0, status -> 1,name->"govind singh")

A: If you use immutable maps, you can use the - method to create a new map without the given key:
val mx = Map("data" -> "sumi", "rel" -> 2, "privacy" -> 0)

val m = mx("privacy") match {
    case 0 => mx - "data"
    case _ => mx
}

=> m: scala.collection.immutable.Map[String,Any] = Map(rel -> 2, privacy -> 0)

If you use mutable maps, you can just remove a key with either -= or remove.

A: If you're looking to scale this up and remove multiple members, then filterKeys is your best bet:
val a = Map(
  "data"    -> "sumi",
  "rel"     -> "2",
  "privacy" -> "0",
  "status"  -> "1",
  "name"    -> "govind singh"
)

val b = a.filterKeys(_ != "data")

A: That depends on the type of Scala.collection Map you are using. Scala comes with both mutable and immutable Maps. Checks these links:
http://www.scala-lang.org/api/2.10.2/index.html#scala.collection.immutable.Map
and
http://www.scala-lang.org/api/2.10.2/index.html#scala.collection.mutable.Map
In both types of maps, - is usually the operation to remove a key. The details depend on the type of map. A mutable map can be modified in place by using -=. Something like
if (m.contains("privacy") && m.getOrElse("privacy", 1) == 0) {
    m -= "play"
}

On the other hand, an immutable map can not be modified in place and has to return a new map after removing an element.
if (m.contains("privacy") && m.getOrElse("privacy", 1) == 0) {
    val newM = m - "play"
}

Notice that you are creating a new immutable map.

A: val m = Map("data" -> "sumi", "rel" -> 2, "privacy" -> 0,"status" -> 1,"name"->"govind singh")
scala> if(m("privacy")==0) m.filterKeys(_ != "data")
res63: Any = Map(name -> govind singh, rel -> 2, privacy -> 0, status -> 1)

redpajama_set_name: RedPajamaStackExchange


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Do not use with Aqueous or Acrylic paints.


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A Although willpower does not shape our decisions, it determines whether and how long we can follow through on them. It almost single-handedly determines life outcomes. Interestingly, research suggests the general population is indeed aware of how essential willpower is to their wellbeing; survey participants routinely identify a 'lack of willpower' as the major impediment to making beneficial life changes. There are, however, misunderstandings surrounding the nature of willpower and how we can acquire more of it. There is a widespread misperception, for example, that increased leisure time would lead to subsequent increases in willpower.
B Although the concept of willpower is often explained through single-word terms, such as 'resolve' or 'drive', it refers in fact to a variety of behaviours and situations. There is a common perception that willpower entails resisting some kind of a 'treat', such as a sugary drink or a lazy morning in bed, in favour of decisions that we know are better for us, such as drinking water or going to the gym. Of course this is a familiar phenomenon for all. Yet willpower also involves elements such as overriding negative thought processes, biting your tongue in social situations, or persevering through a difficult activity. At the heart of any exercise of willpower, however, is the notion of 'delayed gratification', which involves resisting immediate satisfaction for a course that will yield greater or more permanent satisfaction in the long run.
C Scientists are making general investigations into why some individuals are better able than others to delay gratification and thus employ their willpower, but the genetic or environmental origins of this ability remain a mystery for now. Some groups who are particularly vulnerable to reduced willpower capacity, such as those with addictive personalities, may claim a biological origin for their problems. What is clear is that levels of willpower typically remain consistent over time (studies tracking individuals from early childhood to their adult years demonstrate a remarkable consistency in willpower abilities). In the short term, however, our ability to draw on willpower can fluctuate dramatically due to factors such as fatigue, diet and stress. Indeed, research by Matthew Gailliot suggests that willpower, even in the absence of physical activity, both requires and drains blood glucose levels, suggesting that willpower operates more or less like a 'muscle', and, like a muscle, requires fuel for optimum functioning.
D These observations lead to an important question: if the strength of our willpower at the age of thirty-five is somehow pegged to our ability at the age of four, are all efforts to improve our willpower certain to prove futile? According to newer research, this is not necessarily the case. Gregory M. Walton, for example, found that a single verbal cue – telling research participants how strenuous mental tasks could 'energise' them for further challenging activities – made a profound difference in terms of how much willpower participants could draw upon to complete the activity. Just as our willpower is easily drained by negative influences, it appears that willpower can also be boosted by other prompts, such as encouragement or optimistic self-talk.
E Strengthening willpower thus relies on a two-pronged approach: reducing negative influences and improving positive ones. One of the most popular and effective methods simply involves avoiding willpower depletion triggers, and is based on the old adage, 'out of sight, out of mind'. In one study, workers who kept a bowl of enticing candy on their desks were far more likely to indulge than those who placed it in a desk drawer. It also appears that finding sources of motivation from within us may be important. In another study, Mark Muraven found that those who felt compelled by an external authority to exert self-control experienced far greater rates of willpower depletion than those who identified their own reasons for taking a particular course of action. This idea that our mental convictions can influence willpower was borne out by Veronika Job. Her research indicates that those who think that willpower is a finite resource exhaust their supplies of this commodity long before those who do not hold this opinion.
F Willpower is clearly fundamental to our ability to follow through on our decisions but, as psychologist Roy Baumeister has discovered, a lack of willpower may not be the sole impediment every time our good intentions fail to manifest themselves. A critical precursor, he suggests, is motivation – if we are only mildly invested in the change we are trying to make, our efforts are bound to fall short. This may be why so many of us abandon our New Year's Resolutions – if these were actions we really wanted to take, rather than things we felt we ought to be doing, we would probably be doing them already. In addition, Muraven emphasises the value of monitoring progress towards a desired result, such as by using a fitness journal, or keeping a record of savings toward a new purchase. The importance of motivation and monitoring cannot be overstated. Indeed, it appears that, even when our willpower reserves are entirely depleted, motivation alone may be sufficient to keep us on the course we originally chose.
27 Willpower is the most significant factor in determining success in life.
28 People with more free time typically have better willpower.
29 Willpower mostly applies to matters of diet and exercise.
31 Researchers have studied the genetic basis of willpower.
32 Levels of willpower usually stay the same throughout our lives.
33 Regular physical exercise improves our willpower ability.
Look at the following statements (Questions 37–40) and the list of researchers below.
Match each statement with the correct person, A–D.
34 identified a key factor that is necessary for willpower to function.
35 suggested that willpower is affected by our beliefs.
36 examined how our body responds to the use of willpower.
37 discovered how important it is to make and track goals.
38 found that taking actions to please others decreases our willpower.
39 found that willpower can increase through simple positive thoughts.
Which of the following is NOT mentioned as a factor in willpower?


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Eighty-five enterprises from 21 Chinese cities and 20 U.S. cities gathered for the second U.S.-China Cities Forum on Economic Cooperation and Investment that runs from Friday to Saturday.
A total of 42 investment projects were involved, covering 21 sectors -- including manufacturing, energy conservation and environmental protection, electronic information, chemical and pharmaceutical.
China's Finance Minister Xie Xuren said at the opening ceremony that China and the United States share different resource advantages, and strongly complement each other in economic development.
The two economies should expand cooperation in more sectors, especially in energy and environmental protection, biomedicine and infrastructure, so as to turn opportunities into tangible results, Xie said.
The forum, jointly sponsored by Chinese and U.S. finance ministries, this year seeks to form contracts on the basis of spoken agreements reached between China and U.S. top officials over the past year, including Chinese President Hu Jintao's visit to the United States in January 2011.
The forum was first held in Seattle on April 19 and 20 last year.
China and the United States are each's second largest trading partner. Bilateral trade currently totals 446.6 billion U.S. dollars.


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If you have provided a valid email address with your order, we'll send an email to you to let you know your return has been processed. Please call for RA# to be issued, and we will provide you with mailing address information.
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Due to the nature of the hair goods industry, we find it necessary to charge a 20% re-stocking fee. (For example if you made a purchase of $60.00 plus tax, but then changed your mind after receiving the order, there would be a $12.00 re-stocking charge). We reserve the right to increase the re-stocking fee for customers who make frequent returns. Such customers will be given advance notice of their fee status.
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text:
National Forum
CPA threatens strike action
Club Players Association executive member Aaron Kernan, Secretary Declan Brennan and Chairman Michael Briody
The Club Players' Association (CPA) has raised the prospect of its membership going on strike if a solution to the "fixtures crisis" isn't found soon.
"Over two years have passed since the CPA was established to help 'Fix The Fixtures'," the players' body noted in a statement.
"Unfortunately, the fixtures crisis remains, and nothing has been done by GAA management to materially address the situation or achieve real change.
"Over the last number of months, we have been in extensive dialogue with the Director General and An Uachtarán of the GAA. Substantial and constructive breakthroughs were made in December and January, whereby we got firm commitments that a Fixtures Group would be set up to look at all competitions. It would have an independent chairperson, with representatives from all key stakeholders, including the CPA.
"We have a clear trail of correspondence from Croke Park about the new Fixtures Group, but it has yet to be constituted or a date set for meetings. We thought we were being listened to on the formation of the group, its time-frame of activity and its terms of reference. Sadly, progress has since ground to a halt, despite our efforts and enthusiasm. Words but no action.
"The GAA recently published its new manifesto, which speaks to a lot of what we believe in as members.
"Sadly in the case of fixtures, some of the sentiments ring hollow, and conversely some ring startlingly true. Uncomfortably and perhaps unintentionally so for the manifesto's authors. So we are now looking for action, not words."
The statement concluded: "We will be in touch again soon to update you on next steps, including possible escalation. We ask you as a club player to get ready to stand up for your club against continued inaction."

redpajama_set_name: RedPajamaCommonCrawl


Shohei Ohtani hit his second homer in two games at Angel Stadium in the fifth inning, and Zack Cozart had walk-off homer in the 13th inning of the Los Angeles Angels' 3-2 victory over the Cleveland Indians on Wednesday.
One day after Ohtani hit an electrifying three-run homer in his first home plate appearance, the Japanese two-way rookie connected for a tying two-run homer off AL Cy Young winner Corey Kluber in his second game at the Big A. Ohtani also singled leading off the 10th to go 2 for 5, giving him six hits this season and five in his only two home games.
The leadoff hitter extraordinaire with the ragged beard and impressive mullet will be patrolling the Colorado Rockies outfield for quite some time. NL batting champion Charlie Blackmon made a long-term commitment rather than test the free-agent market next offseason, agreeing to a contract that guarantees him $108 million over six seasons.
The New England Patriots have signed free-agent tight end Troy Niklas to a one-year deal.
The 6-foot-7, 270-pound Niklas played the last four seasons for the Arizona Cardinals. A 2014 second-round pick out of Notre Dame, Niklas, 25, caught 11 passes for 132 yards last season.
Robert Griffin III is poised to resume his NFL career about 35 miles away from where he started it. The former Washington Redskins quarterback has reached agreement on a 1-year contract with the Baltimore Ravens, who love the idea of having the 2011 Heisman Trophy winner backing up 10-year starter Joe Flacco.
Boston College guards Jerome Robinson and Ky Bowman will enter the NBA draft but won't sign with an agent in case they decide to return to school.
The two players would maintain their eligibility if they are not happy with where they are drafted. Robinson, a junior, was the leading scorer in the Atlantic Coast Conference last season with an average of 20.7 points. Bowman, a sophomore, was an All-ACC honorable mention after averaging 17.6 points, 6.8 rebounds, and 4.7 assists.
The University of Rhode Island has promoted David Cox to men's basketball coach. Cox served as associate head coach for the last four years under Dan Hurley, who left last month to become head coach at UConn.
Conor McGregor's run as UFC lightweight champion is set to end.
The long-expected move that McGregor would lose his 155-belt was officially announced by UFC President Dana White. McGregor has not fought for UFC since he won the 155-pound title at UFC 205 in November 2016. McGregor last fought at all in a boxing match he lost to Floyd Mayweather Jr. on Aug. 26, 2017.


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Northamptonshire Chairman Gavin Warren says that 'honesty' has been the buzzword around the County Ground this winter, and that the Board of Directors has confidence in the coaching staff and playing group to take the club back to the lofty heights it has enjoyed regularly this decade.
The 2019 Specsavers County Championship campaign is fewer than two weeks away, with the Royal London One-Day Cup also beginning within the next month, and Warren has been encouraged by what he has seen over the winter's training.
"The key word this winter has been 'honesty'," he says. "The players sat down with each other at the end of last season and said that 2018 was not acceptable and not at the level they wanted to play at.
"They're professionals and want to play at a high level and have come up with a code of conduct that was presented to the Board, and while we'll have to wait and see what the outcome is in terms of performance the preparation, the fitness levels, the recruitment, everything they set out in that paper has been achieved.
"No balls have been bowled or faced yet, so we'll see how successful that plan is."
Despite the County seeing relative under-performance in 2018 compared to the success experienced in both the Vitality Blast and Specsavers County Championship in the preceding five years Warren says that there was no hesitation in continuing to back Head Coach David Ripley and his staff as they brought fresh thinking to their approach to the new season.
"We believe in our coaching set-up," he added. "Rips has been with us a few years now, and it can be easy to get rid of someone if they have a bad season, as we see in other sports.
"We don't think that's the case at all, we really back Rips, and we've encouraged him to recruit who he wants and have given him some more money as well, and we will back them all the way.
"He deserves it because of what he's achieved, winning the T20 twice and reaching the final, as well as promotion in 2013, and they're going nowhere. They know what they've got to do and are empowered to do it."
Warren also says that a key focus of Northamptonshire is to be a hub for cricket across the region, not just in Northampton and the immediate surrounding area, commenting that "it is key to our future.
"Our role as a County is to take cricket to the surrounding counties and we're doing a lot of work in Luton, Bedfordshire, Huntingdonshire, all around, to try and engage in that community. That's also where we pick players up from. It's important that we connect with people there."
The 2019 Specsavers County Championship campaign begins against Middlesex on Friday, April 5th. Season Tickets remain available, with supporters able to watch all 41 days of scheduled cricket from just £6.50 per day. For full details about Season Ticket options and booking information head to www.northantscricket.com/season-tickets


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No person is an island. We need others to reach success!
A spectre is haunting Europe, the spectre of self-improvement. How many steps have I walked today? How many calories have I burned or eaten? How long were my deep-sleep phases last night? How many likes did I get for my last run? Self-improvement is everywhere and more and more people aim to optimise all spheres of their lives.
If you Google ai???how to become the best version of myselfai??? you get 300 million answers with endless livehacks, tips, strategies, to-dos and not-to-dos. An entire industry has been growing around the inherent desire of self-improvement. It is about selling tools, software, dietary supplements, books, coaching, personal training and ultimately the dream to become the best version of yourself ai??i?? fitter, healthier, smarter, more attractive and successful!
Today technologies permit us to track our life and analyse nearly all aspects of it. Life gets quantified and measured in meters, hours, calories ai??i?? we set goals for ourselves and try to achieve them every day. Suddenly, the subject turns into an object. Monitoring your own progress can be very motivating or even addictive. Smartphones, watches or other ai???wearablesai??i?? help us to understand how much we sleep, walk, cycle, read or eat, becoming our watch-dogs, personal trainers and 24-hour coaches. This provides new health-tracking opportunities, an unprecedented boost and new spin to the recurrent desire of self-improvement.
The increased use of self-optimisation tools can be seen both as a symptom and accelerator of more competitive societies. In the words of the author Michel Houellebecq, what we see happening is the ai???extension of the area of struggle" to different interlinked areas of life: profession, friendship, sexuality, health, culture etc. For example, if I am fitter and more attractive I might also get a better job!
However, does the continuous struggle to optimise ourselves make us happier people with more fulfilled lives? What is the ultimate stimulus for our persistent ai???runai??? ai??i?? is it to do something good for ourselves, or rather to please or become better than others?
Self-optimisation can lead us down a wrong path, which is to believe that we can be self-sufficient, achieve our goals and grow on our own.
My experience living with mentally and physically handicapped people taught me that personal development is not mainly about becoming self-sufficient or autonomous in a narrow sense. My mates with down syndrome were not capable to learn how to write or read properly, however, they had immense emotional intelligence. They taught me that often it is much harder and much more important not to just do things on your own but to assume your vulnerability, accept your limits and sometimes ask for help. Growth and development is not predominantly about achieving progress yourself, but doing it together ai??i?? in all areas. Efficient solutions to complex challenges are always collective, building on team-work, mutual respect, trust and support: multilateralism overcoming Trump's ai???America firstai???, young people engaging in youth organisations to prevent social exclusion, etc.
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We cannot grow or improve just on our own and we have to accept this. We very much need the support of others, their friendship and love. Robinson on his island was definitely very fit and skilled to survive. Nevertheless, his life was also very lonely and sad, making him nearly go completely mad.
Our goal should not be to become millions and billions of Robinsons. Societies full of people who believe they are or can be optimal by themselves are worse case scenarios. Instead, we have to acknowledge our individual imperfections and fully engage in collective improvement. There is so much to do and so many opportunities to build stronger friendships and respect by thinking and behaving more collectively, making our families, friendships, communities and societies better for everyone.
This is not a call against neither trying to become the best version of yourself, nor using technologies to get fitter and healthier. It is a call to be a critical user of these tools and never loose sight of what is essential in life. The essential benchmarks and success indicators are not visible via smartphones and are never only in our own hands ai??i?? they are in the bonds between people and communities. Therefore, letai??i??s not get too much into our own bubbles. Letai??i??s open up and engage with others. Here is my personal story on this matter: I used to plank regularly for a few minutes at my job in the Youth Forum by myself at a corner of the office, because it was so good for my back, neck and mind. Then, one day came the idea and courage to make this a collective moment. Since then everyday at 4 pm everyone in the office who wants, is planking together. It is still good for everyone's heath and mind, but also a great moment for the team, making the office a better place to be and work together.
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Botox bombshell Tamra Barney has been a Real Housewife of Orange County since the beginning....with one minor technicality. This "housewife" is now on the market, having been divorced from ex-husband Simon Barney since 2010. Fans of the Real Housewives love to hate "mean girl" Tamra, whose ongoing catfight with castmate Gretchen Rossi is legendary for its ferocity. Rowr! Tamra's on the up and up in the sixth season of Housewives, moving into her own divorcee digs and hanging out with gal pal Fernanda Rocha. But scandal is never far behind Tamra, and rumors are flying that she and bisexual Brazilian Fernanda are more breast friends than best friends. Also on the up and up this season are male fans, who got a peek at Tamra's California oranges when she let a nip slip in the (televised) bath. Mr. Skin would love to give Tamra's farm-fresh pair a squeeze!


The 42,000-square-foot Tuacahn Center for the Arts located in Utah's Padre Canyon recently updated the aging L-Acoustics dV-Dosc sound system at its 1,920-seat outdoor amphitheater, opting for a new L-Acoustics Kara line array system installed by Poll Sound.
Ivins, UT (September 15, 2016)—The 42,000-square-foot Tuacahn Center for the Arts located in Utah's Padre Canyon recently updated the aging L-Acoustics dV-Dosc sound system at its 1,920-seat outdoor amphitheater, opting for a new L-Acoustics Kara line array system installed by Poll Sound.
The new rig consists of an LCR three-hang array of 11 Kara boxes each, as well as four SB18 subs flown two per side outboard of the left and right main hangs. Additionally, there's 14 coaxial X8 enclosures used for low-front fills and stage monitors, and a dozen larger X12 enclosures: eight facing forward from the rear of the stage and four used for on-stage fills. The entire system is powered by six LA8 and six LA4X amplified controllers.
"The dV-Dosc sound system worked incredibly well here for 10 years; it was subjected to intense heat during the days and rain and other elements year-round, but it never let them down," recalls Deward Timothy, Poll Sound's owner and president, referring to temperatures that routinely hit 105 degrees Fahrenheit during the day. "We recommended the Kara as the best choice for a new system to follow that.


Want to get the latest news on "Courier," Microsoft's rumored-but-not-confirmed answer to Apple's iPad? Microsoft doesn't want to talk about it. But Redmond does have a suggestion for you: Read the gadget blogs.
Do you already know everything about Project Natal and the Cloud? Is Blaise Aguera y Arcas' jaw-dropping TED talk on augmented-reality Bing Maps and Photosynth last month's news? Then check out some of the online chatter surrounding new releases of Window Phone 7 series handsets, Internet Explorer 9 and the upcoming Courier digital journal.
The blog hyperlinks "Courier digital journal" to this Engadget post offering "exclusive pictures and details" about the device. There's a video too, which you can see at the bottom of this post. Looks pretty cool.
Is it the real thing? Device companies and gadget blogs play a weird game about new product releases, and the rules change depending on the company. Apple (AAPL), notably, gives out almost no information to any outlet about what it's up to, right up until the release of whatever it's working on. Other companies beg the blogs to promote their stuff. And a middle tier of companies promote their stuff via surreptitious "leaks" they're happy to see published.
So maybe this is Microsoft's way of putting its seal of approval on something it can't officially approve. On the other hand, we should note that the company's job site is written by a group of Microsoft recruiters, who may not be fully looped in to the company's device plans and/or communications strategy.
Maybe we'll find out. I've asked the company for comment and will let you know if it has anything to say.
UPDATE: That was quick. Microsoft is sticking with its long-running "no comment" stance.


If you're looking to capture this heart-breaking (or cathartic) process in your story, either with your main characters or with those memorable secondary characters (your MC's parents, best-friends or children), then consider these four predictors of relationship breakdown (they are a wonderful way to capture 'show, don't tell' and to create some interesting moments into your story).
Are you seeing how you could weave these behaviours into your own narrative? I hope so! I'd love how you've already done this, or how you plan on showing their unravelling relationship. Comments and feedback are always appreciated. Connecting with others is why I write. You can comment below, or connect with me on Facebook, or Twitter.
Next Post: Is Your Character Really in Love?
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text:
USS Ronald Reagan Enters 7th Fleet Area of Operations
From Mass Communication Specialist 3rd Class Nathan Burke
The U.S. Navy's only forward-deployed aircraft carrier arrived in the 7th Fleet area of operations, Sept. 17.
PACIFIC OCEAN - The U.S. Navy's only forward-deployed Nimitz-class aircraft carrier USS Ronald Reagan (CVN 76) entered the 7th Fleet area of operations, Sept. 17.
"Ronald Reagan's forward deployment to 7th Fleet demonstrates the U.S. commitment to the region and our allies," said Rear Adm. John Alexander, commander, Battle Force 7th Fleet. "It reiterates the importance of the U.S. Navy and Japan Maritime Self-Defense Force interoperability, and ensures we have the most capable platform forward when it matters, where it matters."
The 7th Fleet area of operations includes 36 maritime countries and the world's five largest foreign armed forces – the People's Republic of China, Russia, India, North Korea and the Republic of Korea. Five of the seven U.S. Mutual Defense Treaties are with countries in the area – Republic of the Philippines, Australia and New Zealand, Republic of Korea, Japan, and Thailand.
"It's an important area for us to protect," said Aviation Boatswain's Mate (Handling) 2nd Class Albert Mendez. "Being forward deployed naval forces we're first responders, so we have to be ready at all times."
Ronald Reagan is one of the most modern aircraft carriers. The technology Ronald Regan is bringing to 7th Fleet is an example of the U.S.'s commitment to the region and the progress of bringing the newest equipment to operate with the forward deployed naval forces.
The crew has undergone multiple training evolutions and qualifications to include damage control and medical drills as well as flight deck certifications during the ship's transit from San Diego to prepare for operations in 7th Fleet.
"We are here to defend our nation," said Lt. Cmdr. Freddie Koonce, Ronald Reagan's Aircraft Handling Officer. "In order to prepare for our mission, we have to train. That's how we get better."
Ronald Reagan and its embarked air wing, Carrier Air Wing (CVW) 5, provide a combat-ready force that protects and defends the collective maritime interests of the U.S. and its allies and partners in the Indo-Asia-Pacific region.