Interplay between cell height variations and planar pulsations in epithelial monolayers

Abstract

Biological tissues change their shapes through collective interactions of cells. This coordination sets length and time scales for dynamics where precision is essential, in particular during morphogenetic events. However, how these scales emerge remains unclear. Here, we address this question using the pulsatile domains observed in confluent epithelial MDCK monolayers where cells exhibit synchronous contraction and extension cycles of $\approx 5$ h duration and $\approx 200~ \upmu \textrm{m}$ length scale. We report that the monolayer thickness changes gradually in space and time by more than twofold in order to counterbalance the contraction and extension of the incompressible cytoplasm. We recapitulate these pulsatile dynamics using a continuum model and show that incorporation of cell stiffness dependent height variations is critical both for generating temporal pulsations and establishing the domain size. We propose that this feedback between height and mechanics could be important in coordinating the length scales of tissue dynamics.

Graphic abstract

Appendices

Appendix A: Derivation of equations for the continuum model and analytical estimates

Here, we systematically develop the continuum model described in the main paper and also provide detailed derivations of the relevant equations.

The theory has 2D cell area a, velocity field of the cells $v_i$, internal stress $\sigma _{ij}$ and the contractile matter $\zeta$ as the field variables [34]. The relevant equations are

$$\begin{aligned} {1 \over a}\left( {\partial a \over \partial t} + v_k\partial _k a\right)&= \partial _i v_i, \end{aligned}$$

(A.1)

$$\begin{aligned} \partial _j \sigma _{ij}&= \alpha v_i, \end{aligned}$$

(A.2)

$$\begin{aligned} \sigma _{ij}&= \left[ K {a - a_0 \over a_0} - K_h \varDelta {a - a_0 \over a_0}\right. \nonumber \\ {}&\left. \quad + \zeta + \mu \partial _k v_k \right] \delta _{ij}, \end{aligned}$$

(A.3)

$$\begin{aligned} {\partial \zeta \over \partial t} + \partial _k v_k \zeta&= -{1 \over \tau }(\zeta - \zeta _0) - \beta {a - a_0 \over a_0} + \beta _c \xi _c. \end{aligned}$$

(A.4)

In the above equations, $\alpha$ is the friction coefficient between the monolayer and the substrate, K is the linear area modulus of the cells, $K_h$ is the height modulus of the tissue that is modeled to depend on the area gradient in the tissue (see “Appendix C”). In Eq. A.4, $\tau$ is the time required for myosin to reach its homeostatic value $\zeta _0$, $\beta$ is the coupling coefficient that relates cellular area with $\zeta$ modulation, and $\xi _c$ is time-correlated noise satisfying

$$\begin{aligned}&\langle \xi _c\rangle = 0, \end{aligned}$$

(A.5)

$$\begin{aligned}&\langle \xi _c(x, y, t) \xi _c(x', y', t')\rangle \nonumber \\&\quad = {1 \over 2\tau _c}\exp \left( -{|t-t'| \over \tau _c}\right) \delta (x-x')\delta (y-y'). \end{aligned}$$

(A.6)

In the limit when the correlation time $\tau _c \rightarrow 0$, the noise becomes uncorrelated or white noise. The parameters in the continuum equations and their units in SI are summarized in Table 1—see also in “Appendix D.”

Table 1 SI units of variables and parameters associated with the model

For small perturbations, $\delta a = a - a_0$ and $\delta \zeta = \zeta - \zeta _0$, eliminating the velocity term and linearizing in $\delta a$ and $\delta \zeta$, we get

$$\begin{aligned} (\alpha - \mu \varDelta ) \partial _t {\delta a \over a_0}&= (K \varDelta - K_h \varDelta \varDelta ){\delta a \over a_0} + \varDelta \delta \zeta , \end{aligned}$$

(A.7)

$$\begin{aligned} \partial _t {\delta \zeta \over \zeta _0}&= -{1 \over \tau } {\delta \zeta \over \zeta _0} - {\beta \over \zeta _0}{\delta a \over a_0}. \end{aligned}$$

(A.8)

Here, we neglect the advection term of the form $\zeta _0 \partial _k v_k$ for conceptual simplicity in the weak advection limit. Choosing, ${\bar{\tau }} = {\zeta _0 \over \beta }$ as a reference time scale and $\ell = {\sqrt{K\zeta _0 \over \beta \alpha }}$ as the length scale, we get the following non-dimensionalized dynamical equations.

$$\begin{aligned} \left( 1 - \mu \varDelta \right) \partial _t {\delta a \over a_0}&= (\varDelta - K_h\varDelta \varDelta ){\delta a \over a_0} + {1\over K}\varDelta {\zeta \over \zeta _0}, \end{aligned}$$

(A.9)

$$\begin{aligned} \partial _t {\delta \zeta \over \zeta _0}&= -{\zeta _0 \over \tau \beta } {\delta \zeta \over a_0} - {\delta a \over a_0} + {\beta _c \over \beta }\xi _c. \end{aligned}$$

(A.10)

We can further rewrite these equations to give

$$\begin{aligned} \left( 1 - {\mu } \varDelta \right) \partial _t \left[ {\beta \over \beta _c}{\delta a \over a_0}\right]&= (\varDelta - K_h\varDelta \varDelta )\left[ {\beta \over \beta _c}{\delta a \over a_0}\right] + {1 \over K}\varDelta \left[ {\beta \over \beta _c}{\zeta \over \zeta _0}\right] , \end{aligned}$$

(A.11)

$$\begin{aligned} \partial _t \left[ {\beta \over \beta _c}{\delta \zeta \over \zeta _0}\right]&= -{\zeta _0 \over \tau \beta } \left[ {\beta \over \beta _c}{\delta \zeta \over \zeta _0}\right] - \left[ {\beta \over \beta _c}{\delta a \over a_0}\right] + \xi _c. \end{aligned}$$

(A.12)

Finally, we present the governing equations as

$$\begin{aligned} (1 - \mu \varDelta )\partial _t a&= (\varDelta - K_h\varDelta \varDelta )a + {1 \over K}\varDelta \zeta , \end{aligned}$$

(A.13)

$$\begin{aligned} \partial _t \zeta&= -{1 \over \tau }\zeta - a + \xi _c, \end{aligned}$$

(A.14)

where the non-dimensional dynamical variables are written as $a \equiv \left[ {\beta \over \beta _c}{\delta a \over a_0}\right]$ and $\zeta \equiv \left[ {\beta \over \beta _c}{\delta \zeta \over \zeta _0}\right]$. The four, free, non-dimensional parameters in these equations are $\mu \equiv {\mu \over \alpha \ell ^2} = {\mu \beta \over K\zeta _0}$, $K_h \equiv {K_h {\bar{\tau }} \over \alpha K \ell ^4} = {K_h \alpha \beta \over \zeta _0 K^2}$, ${1 \over K} \equiv {\zeta _0 \over K}$ and $\tau \equiv {\tau \beta \over \zeta _0}$, in terms of the original dimensional parameters. The correlation time of the colored noise $\xi _c$ is written as $\tau _c \equiv {\tau _c \beta \over \zeta _0}$.

Taking Fourier transform of the two coupled equations in space ($x \rightarrow q_x$ and $y \rightarrow q_y$) and time ($t\rightarrow \omega$), we get

$$\begin{aligned} \underbrace{ \begin{bmatrix} -i\omega (1 + \mu q^2) + q^2 + K_h q^4&{} { q^2\over K}\\ 1&{} -i\omega + {1 \over \tau } \end{bmatrix} }_{{{\textbf{C}}}} \begin{bmatrix} {\tilde{a}}\\ {\tilde{\zeta }} \end{bmatrix} = \begin{bmatrix} 0\\ {\tilde{\xi }}_c \end{bmatrix},\nonumber \\ \end{aligned}$$

(A.15)

where ${\tilde{a}}$ and ${\tilde{\zeta }}$ are the space-time Fourier transforms, respectively, of a and $\zeta$, and $q^2 = q_x^2 + q_y^2$. Inverting the ${{\textbf{C}}}$ matrix, we get

$$\begin{aligned} \begin{bmatrix} {\tilde{a}}\\ {\tilde{\zeta }} \end{bmatrix} = {{\textbf{C}}}^{-1} \begin{bmatrix} 0\\ {\tilde{\xi }}_c \end{bmatrix} \end{aligned}$$

(A.16)

The power spectrum from ${\tilde{a}}$ and ${\tilde{\zeta }}$ then becomes

$$\begin{aligned} S_{a}(q_x, q_y,\omega )&= |{\tilde{a}}|^2, \end{aligned}$$

(A.17)

$$\begin{aligned} S_{\zeta }(q_x, q_y, \omega )&= |{\tilde{\zeta }}|^2. \end{aligned}$$

(A.18)

We note that since $q = \sqrt{q_x^2 + q_y^2}$, the power spectrum only in terms of q can now be obtained as $S_a(q, \omega ) = 2\pi q \times S_a(q_x, q_y, \omega )$. Since the linearized mass balance equation Eq. A.1 in terms of the non-dimensionalized variables and parameters is

$$\begin{aligned} \partial _t a = {\beta \over \beta _c} \nabla \cdot {{\textbf{v}}}, \end{aligned}$$

the power spectrum for velocity divergence becomes, $S_d(q, \omega ) = {\beta _c^2 \over \beta ^2}\omega ^2 S_{a}(q, \omega )$. Since we have taken $\xi _c$ to be correlated noise (Eq. A.4), its power spectrum $|\tilde{\xi _c}|^2 = 1/(1 + \tau _c^2\omega ^2)$. Hence, the effective power spectrum for the velocity divergence is given as

$$\begin{aligned} S_d(q, \omega ) = {q^5 \tau ^2 \omega ^2 \over D}, \end{aligned}$$

(A.19)

where

$$\begin{aligned} D&= {\beta ^2 \over 2 \pi \beta _c^2}(1 + \tau _c^2 \omega ^2)(q^4\tau ^2 - 2Kq^2\tau \nonumber \\&\quad \times \, [q^2 + K_hq^4-(1+q^2\mu )\tau \omega ^2] \nonumber \\&\quad +\, K^2[1+\tau ^2\omega ^2]\nonumber \\&\quad \times \, [q^4(1+K_hq^2)^2 + \omega ^2(1 + q^2\mu )^2]). \end{aligned}$$

(A.20)

We can perform a scaling near $q \rightarrow \infty$

$$\begin{aligned} S_d(q\rightarrow \infty , \omega )= {2 \pi \tau ^2 \omega ^2 \over K^2 K_h^2 q^3(1 + \tau ^2 \omega ^2)(1 + \tau _c^2 \omega ^2)}.\nonumber \\ \end{aligned}$$

(A.21)

Upon integrating the $\omega$ terms out, we get

$$\begin{aligned} S_d(q\rightarrow \infty ) = {\pi ^2 \tau \over K^2K_h^2q^3\tau _c(\tau + \tau _c)}, \end{aligned}$$

(A.22)

and near $q \rightarrow 0$ we get

$$\begin{aligned} S_d(q \rightarrow 0, \omega ) = {2 \pi q^5 \tau ^2 \over K^2(1 + \tau ^2 \omega ^2)(1+\tau _c^2 \omega ^2)}; \end{aligned}$$

(A.23)

upon integrating the $\omega$ terms we get

$$\begin{aligned} S_d(q \rightarrow 0) = {\pi ^2 q^5 \tau ^2 \over K^2 (\tau + \tau _c)}. \end{aligned}$$

(A.24)

As derived above, the series expansion for $S_d(q)$ is increasing and decreasing in q for small and large values, respectively, of q. Hence, we could expect the location $q_m$ of the maxima of $S_d(q)$ to approximately be at the intersection of these two curves. Hence, we equate them

$$\begin{aligned} S_d(q\rightarrow \infty ) = S_d(q \rightarrow 0), \end{aligned}$$

(A.25)

to give

$$\begin{aligned} q_m = \left( {1 \over K_h^2 \tau \tau _c}\right) ^{1/8}. \end{aligned}$$

(A.26)

The wavelength associated with this pattern will be given by $l_0 = 2 \pi /q_m$.

Following the same procedure as for q, we now check for the asymptotic behavior for $S_d(q, \omega )$ in terms of $\omega$.

$$\begin{aligned} S_d(q, \omega \rightarrow 0) = {2 \pi q \tau ^2 \omega ^2 \over (K + K K_h q^2 - \tau )^2}. \end{aligned}$$

(A.27)

Similarly,

$$\begin{aligned} S_d(q, \omega \rightarrow \infty ) = {2 \pi q^5 \over K^2(1 + q^2\mu )^2 \tau _c^2\omega ^4}. \end{aligned}$$

(A.28)

We find that the location of maxima for $S_d(\omega )$ is obtained quite well when we equate $S(q, \omega \rightarrow 0)$ and $S(q, \omega \rightarrow \infty )$ corresponding $q = q_m$ as obtained in Eq. A.26. From that, we get that the maxima for $S_d(\omega )$ occur at

$$\begin{aligned} \omega _m = \left( {1 \over \tau \tau _c}\right) ^{{5 \over 12}}\left( \frac{[K - \tau ](\tau \tau _c)^{{1 \over 4}} + K\sqrt{K_h}}{K(K_h^2\tau \tau _c)^{{1 \over 4}} + K\mu }\right) ^{{1 \over 3}} .\nonumber \\ \end{aligned}$$

(A.29)

Based on Eqs. A.21-A.29, we can get some insights into the role the non-dimensionalized parameters $K, K_h, \mu , \tau$ and $\tau _c$ on the behavior of power spectra $S_d(\omega )$ and $S_d(q)$. From Eq. A.26, we find that, as per our model, the three terms $K_h, \tau$ and $\tau _c$ are important for the observation of maxima in $S_d(q)$ as is observed experimentally (Fig. 3e). Hence, the term corresponding to gradients in area in tissue stress (Eqs. 3 and A.3), the active stress homeostatic term (Eqs. 6 and A.14) and the time-correlation component of the noise (Eq. A.6) play key role in setting the length scale of pulsations in the tissue within our model framework. Additionally, from Eqs. A.22 and A.24, we see that the rate at which $S_d(q)$ grows and decays for small and large values, respectively, of q is governed by the parameters $\tau , K$, and $K_h$. From Eq. A.28, it can be seen that although the parameter $\mu$ does not play an essential role in governing the qualitative nature of $S_d(q)$ and $S_d(\omega )$, it modulates the decay behavior of $S_d(\omega )$ for larger values of $\omega$.

In Fig. 3, we plot the normalized power spectrum in terms of the wavenumber q and $\omega$ by defining the following.

$$\begin{aligned} S_d(q)&= {1 \over S_{\textrm{tot}}}\int _{-\infty }^{\infty } S_d(q, \omega )\textrm{d}\omega , \end{aligned}$$

(A.30)

$$\begin{aligned} S_d(\omega )&= {1 \over S_{\textrm{tot}}} \int _{0}^{\infty } S_d(q, \omega )\textrm{d}q, \end{aligned}$$

(A.31)

$$\begin{aligned} S_{\textrm{tot}}&= \int _{-\infty }^{\infty }\int _{0}^{\infty } S_d(q, \omega )\textrm{d}q \textrm{d}\omega . \end{aligned}$$

(A.32)

Note that for notational simplicity, we use the same symbol $S_d$ for the different variants of the power spectrum.

In order to quantitatively obtain the low and high q and $\omega$ behavior of $S_d(q)$ and $S_d(\omega )$, respectively, we show in Fig. 4 log–log plot of the data presented in Fig. 3a, c. Since, as opposed to the theoretical calculation, $S_d(\omega = 0) \ne 0$ for the experimental data, we fit a power law of the form $\omega ^{m}$ while neglecting a few data points at the beginning (Fig. 4a). The exponents $m \approx 1.4$ and $m \approx -1.7$ (Fig. 4a) are not the same as $m \approx 2$ (Eq. A.27) and $m \approx -4$ (Eq. A.28), respectively, as is predicted by the model. Similarly, the exponents $m \approx 1.5$ and $m \approx -4.8$ (Fig. 4b) for $S_d(q)$ also do not match with the theoretical estimate of $m = 5$ (Eq. A.24) and $m = -3$ (Eq. A.22).

Fig. 4

Log–log plot for (a) $S_d(\omega )$ and (b) $S_d(q)$ presented earlier in Fig. 3c, e, respectively. The behavior of the power spectrum at small and large values of $\omega$ and q, respectively, is shown as blue lines in (a, b). a For small values of $\omega$, $S_d(\omega ) \sim \omega ^{1.4}$ and for larger values of $S_d(\omega ) \sim \omega ^{-1.7}$. The theoretical values of these exponents, 2 (Eq. A.27) and $-4$ (Eq. A.28), are not the same as those experimentally observed. b For small values of q, $S_d(q) \sim q^{1.5}$ and for large values of q, $S_d(q) \sim q^{-4.8}$. The theoretical values of these exponents 5 (Eq. A.24) and $-3$ (Eq. A.22) are different than their experimentally obtained counterparts

Appendix B: Power spectrum for cellular pulsations on non-patterned surface

Fig. 5

Power spectra a $S_d(\omega )$, c $S_d(q)$ and the corresponding log–log plots (b) and (d) for pulsatory cellular movements in MDCK epithelial monolayers on non-patterned surfaces. The nature of the power spectra is similar to that obtained for patterned surfaces in Fig. 3c, e. The blue lines and the gray-shaded region in (a) and (c) correspond to the mean and standard deviation, respectively, of $S_d(\omega )$ and $S_d(q)$ obtained over five separate experimental repeats. The power-law fits for low and high values of (b) $\omega$ and (d) q are represented with the blue lines, and the corresponding exponents are indicated

In our previous work [17], we had studied pulsations on epithelial monolayers. Here, similar to the power spectrum $S_d(\omega )$ and $S_d(q)$ for the divergence of pulsatory cellular movements on patterned surfaces discussed in Sect. 3.3 and plotted in Fig. 3c, e, respectively, we do the equivalent for the non-patterned surfaces in Fig. 5a, c. As shown in Fig. 5, the blue curves and the gray-shaded region correspond to the mean and standard deviation of the quantities over five experimental repeats. The peak for $S_d(\omega )$ happens at $\omega /2\pi \approx 0.15~\textrm{h}^{-1}$ corresponding to oscillation period of approximately 6.5 h. Although the peak is not as well defined for $S_d(q)$, its maxima occur at around $q/2\pi \approx 0.004~{ \upmu \textrm{m}}^{-1}$ corresponding to a periodic pattern of approximately $250~{ \upmu \textrm{m}}$. The values are not significantly different from that obtained for the patterned surface. We also check the values of the power spectra at larger and smaller values of $\omega$ (Fig. 5b) and q (Fig. 5d) and obtain the power-law exponents as shown in Fig. 4 for patterned substrate. Although the qualitative rise and decay in q and $\omega$ for the power spectra is similar for the patterned and non-patterned surfaces, the power-law exponents are not the same for these two conditions. However, the exponents for $S_d(q)$ do exhibit a better match with each other. As for the patterned surface, the exponents also do not match the values predicted by the model.

Appendix C: Reasoning behind the term involving $K_h$ in tissue stress

We provide an intuitive reasoning for including the term $K_h \nabla ^2 \left( {a - a_0 \over a_0} \right)$ in tissue stress $\sigma _{ij}$ in Eq. 3. We first note that a cell has at least two modes of deformation. First is the in-plane area change $\delta a = a - a_0$ without any gradients in tissue height h. For an incompressible tissue, $a h = a_0 h_0$ due to which $\delta a/a_0 = - \delta h/h_0$. For the area deformation mode, the energy density of the tissue is $k_1 (a - a_0)^2$. The other mode of deformation in which there is no change in cell area would necessarily involve cell height gradients, and the energy density, to the lowest order, can be written as $k_2 |\nabla h|^2$. Thus, the total energy of the tissue due to these two modes of deformation is

$$\begin{aligned} U = \int _\varOmega [k_1 (a - a_0)^2 + k_2 |\nabla h|^2]\textrm{d}x\textrm{d}y, \end{aligned}$$

(C.33)

over the total region $\varOmega$ of the tissue. We note that the apical and basal layers of the epithelium can have distinct mechanical properties [46]. Since spatial variation in tissue height h can lead to different deformations of the apical and basal layers, the difference in mechanical properties can further contribute to the tissue energy via the $|\nabla h|^2$ term in Eq. C.33. For small deformation of the tissue, the isotropic stress $\sigma$ in the tissue will be obtained by taking the functional derivative of the tissue $\sigma = \delta U/ \delta a$. After noting that $\nabla h = -h_0/a_0\nabla a$, and renormalizing the values of the material coefficients $k_1$ and $k_2$, we get

$$\begin{aligned} \sigma = K \left( {a - a_0 \over a_0} \right) - K_h \nabla ^2 \left( {a - a_0 \over a_0}\right) , \end{aligned}$$

(C.34)

as written in Eq. 3. We note that in the absence of volume incompressibility of the monolayer, we cannot rigorously replace the $\nabla h$ term with $\nabla a$ as described above. Although, in such a case, we do not expect major qualitative changes in our model except for re-scaling of $K_h$, the use of $\nabla a$ term is, strictly speaking, a model approximation.

Appendix D: Estimate for pattern length scale $l_0$

We estimate the typical length $l_0$ given in Eq .10 by proposing the following experimental values for the parameters : substrate friction $\eta \approx 30~{ \textrm{N}\, \textrm{s}\, \textrm{m}^{-1}}$ between the cells and the surface [47, 48], cell stiffness $Y \approx 100~{\textrm{pN} \, \upmu \textrm{m}^{-2}}$ in terms of the typical force per unit area for cell–cell contacts [49], myosin homeostasis time $\tau \approx 60~\textrm{min}$ [23] and myosin fluctuation correlation time along cell–cell junctions $\tau _c \approx 2~\textrm{min}$ [35]. Taking for cell area $a_0 \approx 100~{ \upmu \textrm{m}}^2$ and for cell height $h_0 \approx 10~{ \upmu \textrm{m}}$, we can express $\alpha \approx \eta \times a_0$ and $K_h \approx Y\times a_o\times h_0$. Using these numbers, we obtain an approximate length scale associated with the pulsating pattern $l_0 \approx 100~{ \upmu \textrm{m}}$, which is consistent with the typical length scales measured experimentally.

Supplementary movie caption: Movie 1: Height fluctuations during monolayer pulsations. In the top, domain of MDCK-E-Cadherin-GFP cells on FN grid is shown at two z-planes: 2.5 Â$\mu$m and 8 Â$\mu$m. Scale bar, 100 Â$\mu$m. In the bottom, 3D surface view of cells where height is encoded as intensity is shown. Color bar shows the height values. Time in hh:mm.