Hydraulic and electric control of cell spheroids

The total tissue stress is decomposed as $\sigma_{\alpha\beta}=\sigma^{\rm c}_{\alpha\beta}+\sigma^{\rm f}_{\alpha\beta}$ where here and in the following the superscripts $\rm c,f$ stand for the cells and interstitial fluid contributions, respectively. Neglecting inertia and in the absence of external bulk forces, force balance within the tissue reads:

where $f_{\alpha}$ represents internal forces between the cells and the interstitial fluid, and where summation over repeated indices is implied.

To obtain the dynamics of the spheroid, that is, the time evolution of the inner and outer radii $R_{1}$ and $R_{2}$, we now need to specify the material properties (or constitutive equations) of the tissue. In our coarse-grained description, the interstitial fluid is driven by pressure gradients, while fluid viscosity contributes to the internal forces $f_{\alpha}$ between fluid and cells. We therefore write the fluid stress as $\sigma^{\rm f}_{\alpha\beta}=-P^{\rm f}\delta_{\alpha\beta}$. For the cell stress, the description of the material properties is made easier by decomposing the stress into a traceless symmetric part $\tilde{\sigma}^{\rm c}_{\alpha\beta}$ and an isotropic part $\sigma^{\rm c}\delta_{\alpha\beta}$. The constitutive equation for the isotropic part then reads Sarkar et al. (2019); Duclut et al. (2019):

We emphasize that the cell stress constitutive equations (Eqs. (2a) to (2c)) reflect the effective viscous properties of tissues at long time as a consequence of cellular growth and death. We review the derivation of these constitutive equations in App. A. A key feature of our work is that flows and electric fields do influence cell division and death, and therefore growth and shrinkage of the spheroid Blackiston et al. (2009); Sarkar et al. (2019); Duclut et al. (2019).

The constitutive equation for the internal force density between cells and interstitial fluid including all the linear terms allowed by symmetry can be written as¹¹1Note that additional terms must be added to Eqs. (3) and (4) for a system lacking spherical symmetry or for a system where the polarity is not purely radial with unit norm, see App. A.3 for details.

The first term $-\kappa(v^{\rm c}_{\alpha}\!-\!v^{\rm f}_{\alpha})$ accounts for the friction between the interstitial fluid and the cells and leads to Darcy’s law Darcy (1856) in the description of porous materials. The permeation coefficient can be estimated as $\kappa\simeq\eta^{\rm f}/a^{2}$, where $\eta^{\rm f}$ is the interstitial fluid viscosity and $a$ a typical interstitial distance. The term $\lambda_{1}p_{\alpha}$ accounts for active fluid pumping by the cells, while the terms proportional to $\lambda_{2}$ correspond to the electroosmotic contribution. The last term, proportional to $\lambda_{3}$, characterizes a differential pumping term due to the bending of the cells Ramaswamy et al. (2000).

To complete the tissue properties description, one finally needs to specify the constitutive equation for the electric current density²²2Note that additional terms must be added to Eqs. (3) and (4) for a system lacking spherical symmetry or for a system where the polarity is not purely radial with unit norm, see App. A.3 for details.

where $\Lambda_{2}$ is the electric conductivity of the tissue. The term proportional to $\bar{\kappa}$ characterizes the current due to the (relative) flow of ions between cells as a consequence of a reverse electroosmotic effect Kirby (2013). The coefficient $\Lambda_{1}$ characterizes the contribution of ion pumping to the electric current, while the coefficient $\Lambda_{3}$ is an active flexoelectric coefficient. It indicates that a spatially nonuniform cell polarity orientation is obtained in response to an electric field, and has been shown to play a crucial role in the nucleation of a lumen in spherical cell aggregates Duclut et al. (2019).

The drain can be used to impose an external fluid flow or an external electric current in two equivalent ways: either by imposing directly a volumetric flow rate $Q^{\rm ext}$ and electric current $I^{\rm ext}$ through the drain, or, alternatively, a pressure difference $\Delta P=P^{\rm d}-P^{\rm ext}_{2}$ (with $P^{\rm ext}_{2}$ the pressure at the outer spheroid boundary and $P^{\rm d}$ the pressure at the outer end of the drain, see Fig. 1) and an electric potential difference $\Delta U=U^{\rm d}-U^{\rm ext}_{2}$ (with $U^{\rm ext}_{2}$ the electric potential at outer spheroid boundary and $U^{\rm d}$ the electric potential at the outer end of the drain) can be applied.

In both cases, the imposed external fluid flow and external electric current density at the lumen boundary read:

where $v^{\rm ext}_{1}$ and $j^{\rm ext}_{1}$ are the fluid velocity and electric current density at the boundary between the spheroid and the lumen.

In the following, we focus on the case where $Q^{\rm ext}$ and $I^{\rm ext}$ are imposed externally. The case of a pressure difference or electric potential difference imposed by the drain are discussed in App. C.

If cell density and interstitial fluid density are equal and constant, which we assume in the following, then the total volume flux $v_{\alpha}=\phi v^{\rm c}_{\alpha}+(1-\phi)v^{\rm f}_{\alpha}$ is divergence-free: $\partial_{\alpha}v_{\alpha}=0$ (see App. A for details). In the presence of the drain, which imposes a nonvanishing fluid velocity at the inner boundary of the spheroid, the integration of the total flow incompressibility in spherical coordinates yields a relation between the fluid velocity $v_{r}^{\rm f}$ and cell velocity $v_{r}^{\rm c}$ in the tissue, which reads:

Similarly, charge conservation in the quasistatic limit $\partial_{\alpha}j_{\alpha}=0$ can be integrated in the case where an external current density $j^{\rm ext}_{1}$ is imposed on the inner boundary, yielding the current density $j_{r}=j^{\rm ext}_{1}\left(R_{1}/r\right)^{2}$ throughout the tissue.

The spheroid is surrounded by an external fluid both inside (in the lumen), and outside. This fluid exerts a hydrostatic pressure on the tissue that is balanced by the tissue surface tension and by the total normal stress at the boundaries:

where we have introduced the inner and outer tissue surface tensions $\gamma_{1}$ and $\gamma_{2}$. Fluid exchange between the spheroid and the outside is driven by osmotic conditions:

Here, $K_{1,2}$ are the permeability of the interfaces to water flow. The fluxes $J_{{\rm p},1}$ and $J_{{\rm p},2}$ can be nonzero as a result of active pumps and transporters that maintain an osmotic pressure difference, and act effectively as water pumps. We have also introduced $v^{\rm ext}_{2}$, the external flow imposed at outer spheroid boundary. Conservation of the volumetric flow directly yields $v^{\rm ext}_{2}=v^{\rm ext}_{1}(R_{1}/R_{2})^{2}$.

The normal velocity of the cells at the boundaries has to match the growth of the spheroid radii. An increased cell proliferation in a thin surface layer has been observed in growing spheroids Delarue et al. (2014); Montel et al. (2011); Delarue et al. (2013). We thus allow for a thin surface layer of cells, both facing outside and to the lumen, to have a growth rate that differs from the bulk. The cell velocity boundary conditions then read:

where $v_{i}=\delta k_{i}n^{\rm c}_{i}h/n^{\rm c}$ with $h$ the thickness of the boundary layers, $n^{\rm c}_{i}$ and $\delta k_{i}$ the cell number density and the cell growth rate in the surface layers, respectively.

Using our model and solving numerically Eqs. (38a) and (38b) (see App. D), we can analyse various protocols for the suppression of a spheroid. The dynamics of the spheroid and its lumen, $R_{2}(t)$ and $R_{1}(t)$, is therefore studied for different external flow protocols $Q^{\rm ext}(t)$ and electric current protocols $I^{\rm ext}(t)$. To keep the discussion as general as possible, we introduce dimensionless quantities: a dimensionless radius $r_{2}(\hat{t})=R_{2}(t)/R_{0}$, time $\hat{t}=t/\tau_{0}$, external volumetric flow $\hat{Q}=Q^{\rm ext}/Q_{0}$ and electric current $\hat{I}=I^{\rm ext}/I_{0}$, where we have defined:

Note that the effective pressure $P^{\rm eff}_{2}$ introduced above is a modification of the homeostatic pressure $P^{\rm c}_{\rm h}$ by the external osmotic pressure and by electric and active contributions. Its expression can be found in Table 2 and in App. B.

We display in Fig. 2 different protocols that can be applied to a growing spheroid in order to suppress it. If the external flow magnitude is sufficiently large and if it is applied long enough, the interventions are successful and lead to the spheroid suppression. Figure 2(b) shows an example of a successful suppression of the spheroid as a sufficiently strong flow has been applied until the spheroid is suppressed. A larger flow magnitude leads to a faster spheroid suppression (Fig. 2(c)). We also show in Fig. 2(d) an example of successful suppression protocol for which the magnitude of the flow is lowered as the spheroid size decreases.

Conversely, if the magnitude of the imposed external flow is not sufficient, or if the duration of the flow is too short, the protocol can be unsuccessful in suppressing the spheroid. The spheroid may only have a slower growth rate (Fig. 2(e)) or shrink significantly but resume growing as soon as the external flow is turned off (Fig. 2(f)). Note that for any spheroid size, there exists a critical flow magnitude $|Q_{c}|$ such that imposing a steady flow at magnitude $|Q^{\rm ext}|>|Q_{c}|$ will eventually lead to the spheroid suppression, while $|Q^{\rm ext}|<|Q_{c}|$ will be unsuccessful. Figure 2(b) shows a protocol with $|Q^{\rm ext}|\simeq|Q_{c}|$.

In the examples above we have focused on the case where an external volumetric flow is imposed. The same analysis and similar procedures can be used in the case where an external current $I^{\rm ext}(t)$ is imposed (see Fig. 3). We show in Fig. 3(b) a protocol that leads to the suppression of the spheroid. Note that an increased shrinking is obtained if one applies simultaneously an electric current and an external flow, as both effects are additive. Such intervention is displayed in Fig. 3(c) where we observe a faster shrinking due to external flow in addition to the application of an electric current.

Importantly, we emphasize that the protocols we are discussing here are slow and take place on long time scales: using parameters values displayed in Table 3, the examples shown in Figs. 2 and 3 correspond to $R_{0}\simeq 1$ cm and $\tau_{0}\simeq 10$ days. This shows that the suppression of the spheroid requires a slow and steady flow or the application of a small electric current for several weeks.

We now discuss quantitatively how a drain can be used to control the size of a spheroid. To keep the discussion as simple as possible, we consider here the limit of a spheroid enclosing a small lumen (or lumenless). In the limit of a small lumen compared to spheroid size $R_{1}\ll R_{2}$, the differential equation that describes the dynamics of the spheroid shrinks to (see App. B):

where we use the dimensionless radius $r_{2}$ and time $\hat{t}$ introduced in the previous section. The dimensionless parameters $w_{0,1}$ describe the effects of an externally-imposed flow or electric current. If they are set to zero, Eq. (13) reduces to the dynamics of a lumenless spheroid without drain as studied in Ref. Duclut et al. (2019). They are defined as $w_{n}=j_{n}\hat{I}+u_{n}\hat{Q}$ with $n=0,1$ and where $\hat{Q}$ and $\hat{I}$ are the dimensionless volumetric flow and electric current, respectively, as defined above. The effects of an imposed external flow are captured by the coefficients:

Here, $u_{1}$ describes a permeation effect due to the finite permeability of the spheroid to fluid flows and involves the ratio of the surface permeability $K_{2}$ and of the (effective) bulk permeability $\propto 1/\kappa^{\rm eff}$. The parameter $u_{0}$ includes direct effects of the external flow – proportional to $Q^{\rm ext}/K_{2}$ in the dimensional equation corresponding to $u_{0}=1$ – which reflects the spheroid volume change imposed by the external flow due to tissue incompressibility (see also App. B). We have furthermore defined

which represents the bioelectric and biohydraulic contribution. Indeed, this term is a sum of terms proportional to the parameters $\nu_{i}$, and thus stems from the coupling between the electric field (or the interstitial fluid flow) and the cell polarity that appears in the cell stress.

Imposing an external electric current $I^{\rm ext}$ also contributes to the spheroid growth control via the parameters:

The coefficient $j_{1}=1$ corresponds to electroosmotic flow due to the imposed current. The coefficient $j_{0}$ accounts for bioelectric and biohydraulic contributions where

Note that since the coefficients $\beta_{\rm u,j}$ are a sum of phenomenological parameters for which we only have order-of-magnitude estimates, it is difficult to obtain a reliable estimate of their magnitude and even of their sign. However, estimating upper bounds for $\beta_{\rm u,j}$ suggests that these bioelectric and biohydraulic contributions are small compared to the other effects. A further discussion of these contributions requires precise estimates of the bioelectric and biohydraulic couplings that could be obtained from experiments on spheroids in presence of a drain.

In equation (13), we have used the definitions Duclut et al. (2019)

Here, $\lambda^{\rm eff}$ is an an effective pumping coefficient and $\gamma^{\rm app}_{0,2}$ are apparent tensions defined in App. B. The apparent surface tension $\gamma^{\rm app}_{2}$ is a modification of the tissue surface tension $\gamma_{2}$ stemming from the flexoelectric term proportional to $\Lambda_{3}$. This flexoelectric contribution plays a crucial role in lumen nucleation Duclut et al. (2019). In the dynamics of the outer radius of the spheroid, its effect is however minor. Finally, the effective pumping $\lambda^{\rm eff}$ combines the active pumping $\lambda_{1}$ and an electric contribution $\Lambda_{1}\lambda_{2}/\Lambda_{2}$ due to electroosmosis. The parameters introduced above are summarized for convenience in Table 2, and their corresponding values can be computed using Table 1.

We use Eq. (13) to discuss how external currents and flows can be used to control the growth and shrinkage of a spheroid. Note that whether the spheroid grows (${\rm d}r_{2}/{\rm d}t>0$) or shrinks (${\rm d}r_{2}/{\rm d}t<0$) only depends on the numerator of Eq. (13).

Figure 4 shows phase-space trajectories of the outer radius $r_{2}$ for spheroids that are able to grow in the absence of a drain. Figure 4(a) displays the case without external flow or current. The following scenarios are possible: the spheroid may be growing with an unstable fixed point at $r_{2}=0$ (green curve); alternatively, there might be an additional stable fixed point corresponding to a steady-state, either with finite radius $r_{2}^{*}\neq 0$ (blue curve), or with vanishing radius $r_{2}^{*}$ (orange curve). Imposing an external flow implies nonzero values of $w_{0}$ and $w_{1}$, which are negative in the case where fluid flows out of the lumen (as depicted in Fig. 1(b)). The case of a nonvanishing value of $w_{0}$ is displayed in Fig. 4(b): in this case, the trajectories are shifted downward by an amount $w_{0}$. In this case, even the the growing spheroid (green curve) has a critical radius below which it shrinks (empty green circle). The stable steady-state (blue curve) has moved to $r_{2}^{*}=0$. Figure 4(c) shows the effect of a nonvanishing value of $w_{1}$: the slope of the phase-space trajectories is modified. Note that this perturbation favors shrinking of the spheroid: the unstable critical radius below which the spheroid shrinks (empty circles along the $x$ axis) is shifted to larger values compared to the case without external perturbation shown in Fig. 4(a).

We can estimate the critical volumetric flow $Q_{c}$ needed to change a growing spheroid of size $R_{2}$ to a shrinking one. From Eq. (13), we obtain

where the minus sign indicates that $Q^{\rm ext}<0$ is required to induce shrinkage (corresponding to flow out of the drain, see Fig. 1(b)). We obtain a similar expression for the critical electric current that is required to induce shrinkage:

Note that $\lambda_{2}<0$ and therefore $I_{c}$ is positive (corresponding to an electric current into the drain, see Fig. 1(a)). One can therefore always find a value of the externally applied flow or current to turn a growing spheroid into a shrinking one.

We conclude this section by presenting state diagrams for a spheroid of initial size $R_{2}(t=0)=R_{2}^{0}$ to which a steady external flow $Q^{\rm ext}$ is imposed, see Fig. 5. We consider a spheroid with a drain inserted to the center but without lumen. This corresponds to choosing $R_{1}$ equal to the outer radius of the drain. For simplicity, we fix $R_{1}=2R_{\rm d}$, where $R_{\rm d}$ is the inner drain radius. We again consider spheroids that can grow in the absence of a drain. Figure 5(a) displays the state diagram for the case where the unperturbed spheroid is growing at all radii (corresponding to the green curve in Fig. 4(a)). Figure 5(b) shows the state diagram for a spheroid which is growing above a critical radius (corresponding to the yellow curve in Fig. 4(a)). Figure 5(c) is the state diagram for spheroids that, in the absence of a drain, either reach a steady-state radius or grow without bounds (corresponding to the blue curve in Fig. 4(a)).

Three different regions exist in the diagrams shown in Fig. 5: (i) a growth region (green), where the imposed flow is not sufficient to arrest growth. Note that for increasing values of $Q^{\rm ext}$, growth slows within this region; (ii) a steady-state region (in orange), in which the spheroid reaches a finite size. Within this region a larger imposed flow drives the spheroid to smaller steady-state sizes; (iii) a suppression region (in blue) where the spheroid shrinks until it is suppressed. We define that a spheroid has been suppressed when its size becomes smaller than a cut-off size $R_{2}=R_{1}+h_{\rm c}$, with $h_{\rm c}=10~{}\mu$m a typical cell thickness. In the region of spheroid suppression, the time $t_{\rm s}$ that it takes for the spheroid to be suppressed is indicated by shades of blue. For fixed initial spheroid size, the time for suppression decreases as the magnitude of the imposed flow increases.

The solid black lines indicate boundaries between the different regions. Note that the line separating the steady-state region and the suppression region depends on the cut-off $h_{\rm c}$. The dashed white line indicates the nullcline ${\rm d}r_{2}/{\rm d}\hat{t}=0$ and gives the critical value of the external flow $Q_{c}$ required to switch from a growing to a shrinking spheroid (given by Eq. (19) for $R_{1}=0$). Note that below the nullcline the spheroids grow and above the nullcline they shrink. Note that a similar analysis can be performed in the case of an external current $I^{\rm ext}$. In this case the same behaviors are found: growth, suppression and steady-state regions which depends on the initial spheroid size and on the magnitude of the imposed electric current.

We thank C.T. Lim and T.B. Saw for useful discussions.