A mechano-osmotic model for cellular volume control that integrates mechanical force balance with fluid and ion fluxes

To approach the problem of multicellular volume regulation, we initially consider two cells (Fig. 1) held together via cadherin- and catenin-mediated complexes. As these complexes stabilize on the membrane, connexin structures also assemble and couple with identical units on the neighboring cell to form GJs. These channels connect the cytoplasm of both cells, permitting passive transport of fluid, ions, and small molecules³⁴. GJs typically remain open during their lifecycle, though may close in response to high Ca²⁺ concentrations or low pH which serves to protect the cell from dying neighbors³⁵. Importantly, movement of fluid and ions not only depends on GJ-mediated transport, but also on passive channels and active ion pumps within the cell membrane that permit transfer to and from the extracellular environment.

GJ-mediated ion and fluid transport between cells

To understand how water and ions move between the two cells, we first consider the role of hydrostatic and osmotic pressure, denoted by P and ∏, respectively. A detailed derivation for our thermodynamic framework is provided in Supplementary Notes 1–3, with the key governing equations briefly discussed here. GJs have a diameter in the range of 1.5−2 nm, and can be approximated as fully non-selective to water molecules (diameter \(\approx 0.275\;{\mathrm{{nm}}}\)) and ions (diameters \(\approx 0.1 - 0.2\;{\mathrm{{nm}}}\)). Movement of water between cells is driven by a difference in hydrostatic pressure (see Supplementary Note 2 for motivation) such that the volume flux across GJs (from cell i + 1 to cell i) is described by \(J_{{\mathrm{v}},{\mathrm{g}},i} = - L_{{\mathrm{p}},{\mathrm{g}}}( {P_i - P_{i + 1}} )\), where \(L_{{\mathrm{p}},{\mathrm{g}}}\) is a constant that relates to the water permeability of GPs. We can therefore assume changes in cell volume \(V_{{\mathrm{c}},i}\) are given by:

where A_g is the surface area of the membrane connected to the neighboring cell. GJs also permit a flow of ions between cells, as driven by diffusive and advective flow (see Supplementary Note 2). Assuming dilute conditions, the cell’s internal osmotic pressure relates to the number of ions N_i in the cytosol via Van’t Hoffs equation \({\Pi}_i = N_iRT/V_{{\mathrm{c}},i}\), where R is the gas constant and T is the absolute temperature. The advective/diffusive behavior may then be characterized by an ion flow given by \(\dot{n}_{{\mathrm{g}},i} = - c_{\mathrm{s}}^ \ast L_{{\mathrm{p}},{\mathrm{g}}}( {P_i - P_{i + 1}} ) - \omega _{\mathrm{g}}( {{\Pi}_i - {\Pi}_{i + 1}} )\), where \(\omega _{\mathrm{g}}\) is rate constant and \(c_{\mathrm{s}}^ \ast\) is the mean solute concentration across the two connected cells. Under these conditions, the rate of change in the total number of ions in a cell can be determined:

Mechanics of the cell cortex

As water enters a cell, the increase in fluid volume stretches the cell membrane. The mechanical tension in the membrane is complex, controlled by membrane–cytoskeleton adhesion, cortical stiffness, and active myosin contractility^36,37. We treat the membrane and cortex as a single mechanical structure²⁹, neglecting the possibility of cortical detachment and blebbing. The constitutive law of the cortical structure can be written as \(\sigma _i = \sigma _{{\mathrm{p}},i} + \sigma _{{\mathrm{a}},i}\), where \(\sigma _{{\mathrm{a}},i}\) is the active stress associated with myosin contractility and \(\sigma _{{\mathrm{p}},i}\) is the passive stress predominantly associated with deformation of the actin network (as the actin cortex is much stiffer than the plasma membrane^36,38). With the assumption that the passive stress increases linearly with stretch, it can be expressed as \(\sigma _{{\mathrm{p}},i} = K( {A_{{\mathrm{c}},i}/A_{{\mathrm{c}},i}^0 - 1} )/2\), where K is the effective stiffness, A_c,i the surface area of the cell, and \(A_{{\mathrm{c}},i}^0\) a reference surface area. In addition to internal fluid pressure, the membrane also experiences loading from a spatially uniform external fluid pressure P^ext. Mechanical force balance for a spherical cell with radius r_c,i dictates that the cortical stress can be related to the pressure difference across the membrane \({\Delta} P_i = P_i - P^{{\mathrm{ext}}}\). Therefore, the cortical stress may also be written as \(\sigma _i = {\Delta} P_ir_{{\mathrm{c}},i}/2h_i\), where h_i is the cortical thickness. Further, within a multicellular organoid, proliferation of cells generates compressive solid stresses \(\sigma _{{\mathrm{g}},i}\) that act on neighboring cells³⁹. Deformation of fibrous matrix surrounding the cell cluster compounds the stress, as stretched fibers squeeze on the cluster⁴⁰. Thus, we obtain the following expanded expression for the membrane/cortical stress:

where r₀ is the reference cell radius. Additional details on our treatment of dissipative behavior and model linearization are discussed in Supplementary Note 4.

Fluid and ion exchange with the extracellular environment

In addition to diffusion across GJs, water molecules can move through the semi-permeable cell membrane, enhanced by the presence of aquaporins⁴¹. As these pathways do not permit the diffusion of ions, we can consider them to be fully selective. We assume that the ion concentration in the external media is uniform, such that the external osmotic pressure at any point is given by ∏^ext (non-uniform external ion concentrations are explored in Supplementary Note 9). Therefore, the membrane water flux can be expressed as \(J_{{\mathrm{v}},{\mathrm{m}},i} = - L_{{\mathrm{p}},{\mathrm{m}}}\left( {{\Delta} P_i - {\Delta} {\Pi}_i} \right)\), where \({\Delta} P_i = P_i - P^{{\mathrm{ext}}}\) and \({\Delta} {\Pi}_i = {\Pi}_i - {\Pi}^{{\mathrm{ext}}}\), and \(L_{{\mathrm{p}},{\mathrm{m}}}\) is the permeability coefficient associated with solvent flow through the membrane. Evidently, this flux depends on the difference in osmotic and hydrostatic pressure between the cell and the extracellular environment. We can then extend Eq. 1 to consider this additional water flux such that \({\mathrm{d}}V_{{\mathrm{c}},i}/{\mathrm{d}}t = A_{\mathrm{g}}J_{{\mathrm{v}},{\mathrm{g}},i} + A_iJ_{{\mathrm{v}},{\mathrm{m}},i}\), where \(A_i = 4\pi r_{{\mathrm{c}},i}^2\) is the cell surface area. Note that with our assumption of uniform external hydrostatic and osmotic pressures (e.g. \(P_i^{{\mathrm{ext}}} = P_{i + 1}^{{\mathrm{ext}}} = P^{{\mathrm{ext}}}\)), we can also state the GJ water flux as a function of pressure differences, such that \(J_{{\mathrm{v}},{\mathrm{g}},i} = - L_{{\mathrm{p}},{\mathrm{g}}}\left( {{\Delta} P_i - {\Delta} P_{i - 1}} \right)\). Assuming the cells can be approximated to retain a spherical shape with radius \(r_{{\mathrm{c}},i}\), we achieve the following expanded form for cellular volume change:

The cytosolic ion concentration also depends on exchange with the extracellular environment through selective ion channels. As these channels do not facilitate solvent flow, the associated fluxes assume the general form \(\dot n_i = \omega _j{\Delta} {\Pi}_i\). Mechanosensitive (MS) channels are proteins in the cell membrane that open under a tensile membrane stress⁴² to allow flow of ions from regions where the concentration is high to regions where it is low. As an example, Piezo1 opens under tension to allow a calcium influx, thereby activating Ca²⁺-gated K⁺ channels to relieve osmotic pressure in swollen cells⁴³. The probability of channel opening has been reported to follow a Boltzmann function⁴⁴, and consistent with Jiang and Sun²⁹, we adopt a piecewise linear expression (Fig. 1, yellow curve) to describe the ion flux associated with MS channel permeability \(\dot n_{{\mathrm{ms}},i} = - \omega _{{\mathrm{ms}}}\left( {\sigma _i} \right){\Delta} {\Pi}_i\), such that

where \(\sigma _i\) is the computed cortical stress, \(\sigma _{\mathrm{c}}\) is the threshold stress, below which \(\dot n_{{\mathrm{ms}},i} = 0\), \(\sigma _{\mathrm{s}}\) is the saturating stress, above which the channels are fully open, and β is a rate constant. In addition to these force sensitive channels, there are a number of leak channels (which are always operative) on the membrane² for which we consider a further transmembrane ion flux \(\dot n_{{\mathrm{l}},i} = - \omega _{\mathrm{l}}{\Delta} {\Pi}_i\), where \(\omega _{\mathrm{l}}\) is the associated permeability coefficient.

While the channels described thus far permit passive ion diffusion, there are additional membrane proteins present that actively transport ions against the concentration gradient. These ion pumps require an energy input, such as from ATP hydrolysis, to overcome the energetic barrier associated with moving ions against the concentration gradient. Following Jiang and Sun²⁹, the free energy change associated with pumping action can be expressed as \({\Delta} G = RT\;\log \left( {c_i/c_{{\mathrm{ext}}}} \right) - {\Delta} G_{\mathrm{a}}\), where \({\Delta} G_{\mathrm{a}}\) is an energy input is associated with hydrolysis of ATP. The ion flux associated with active pumping can then be written as \(\dot n_{{\mathrm{p}},i} = \gamma {\prime}{\Delta} G\), where γ′ is a permeation constant. Maintaining our dilute assumption, \({\Delta} G\) can be linearized as \({\Delta} G = RT\left( {{\Pi}_i - {\Pi}^{{\mathrm{ext}}}} \right)/{\Pi}^{{\mathrm{ext}}} - {\Delta} G_{\mathrm{a}}\). We can therefore identify a critical osmotic pressure difference \({\Delta} {\Pi}_{\mathrm{c}}\), determined when \({\Delta} G = 0\), such that \({\Delta} {\Pi}_{\mathrm{c}} = {\Pi}^{{\mathrm{ext}}}{\Delta} G_{\mathrm{a}}/RT\) (noting that when \({\Delta} G \,> \,0\) active pumping is no longer energetically favorable and the pumping direction will reverse⁴⁵). Thus, the ion flux generated by active pumping can be expressed as \(\dot n_{{\mathrm{p}},i} = \gamma \left( {{\Delta} {\Pi}_{\mathrm{c}} - {\Delta} {\Pi}_i} \right)\), where γ is a rate constant. In this framework as we only consider a single ion species, we neglect the influence of electroneutrality and membrane potential. However, a detailed analysis of these additional mechanisms is provided in Supplementary Note 8 where we identify that our simplified approach predicts similar trends to a full electro-osmotic ‘pump-leak’ framework (Supplementary Figs. 5 and 6). Taking pumps and channels into consideration, we can then extend Eq. 2 for a more detailed description of the number of ions within the cell whereby \({\mathrm{d}}N_i/{\mathrm{d}}t = A_{\mathrm{g}}\dot{n} _{{\mathrm{g}},i} + A_i( \dot{n} _{{\mathrm{ms}},i} + \dot{n}_{{\mathrm{l}},i} + \dot{n}_{{\mathrm{p}},i})\). The mean ion concentration between connected cells, \(c_{\mathrm{s}}^ \ast\), can be expressed in terms of osmotic pressure as \(c_{\mathrm{s}}^ \ast = ( {c_i + c_{i + 1}} )/2 = ( {{\Pi}_i + {\Pi}_{i + 1}} )/( {2RT} )\). Based on our established terminology whereby \({\Delta} {\Pi}_i = {\Pi}_i - {\Pi}^{{\mathrm{ext}}}\), this can be rephrased such that \(c_{\mathrm{s}}^ \ast = ( {{\Delta} {\Pi}_i + {\Delta} {\Pi}_{i + 1} + 2{\Pi}^{{\mathrm{ext}}}} )/( {2\;RT} )\). Assuming that \({\Delta} {\Pi}_i/{\Pi}^{{\mathrm{ext}}} \ll 1\), we can therefore approximate \(c_{\mathrm{s}}^ \ast \approx {\Pi}^{{\mathrm{ext}}}/RT\) and the number of cellular ions can be characterized by

Material parameters for all simulations are summarized in Table S1.

GJs amplify differential swelling associated with proliferation-induced solid stresses in neighboring cells

Typically, cell clusters are seeded in a confining matrix, and as the cluster grows it displaces and deforms the elastic matrix, opposing growth and generating solid stresses within the cluster. When cells proliferate within the growing cluster, they push against and apply compressive stresses on their neighbors³⁹, and with the addition of cell adhesion and local jamming this leads to different levels of stress developing spatially. To understand the influence of such solid growth stress on cellular shrinkage and swelling, we first consider the interactions between two connected cells (Fig. 2a). Without loss of generality, we assume there to be a compressive stress \(\sigma _{{\mathrm{g}},1} = \sigma _{{\mathrm{g}},0} + \delta \sigma _{{\mathrm{g}},1}\) acting uniformly on the surface of one cell and for its neighbor to be acted upon by a stress \(\sigma _{{\mathrm{g}},2} = \sigma _{{\mathrm{g}},0}\). For the purpose of illustration, we choose \(\sigma _{{\mathrm{g}},0}\) to equal zero and for \(\delta \sigma _{{\mathrm{g}},1}\) to increase over time to a maximum of 150 Pa (Supplementary Fig. 1a). Varying these parameters will lead to similar trends albeit different magnitudes.

When GJs are active (control case), our model predicts that the loaded cell shrinks and its neighbor swells (Fig. 2b). Initially, in the absence of loading, we find that cells control their volume by regulating their ion concentration through the activities of ion pumps and channels (Fig. 2b and Supplementary Fig. 1). In this unloaded state, MS channels are permeable due to tension in the cell membrane, permitting a constant loss of ions to the external media (Supplementary Fig. 1d). However, active ion pumping ensures there is a continuous ion influx (Supplementary Fig. 1f) for the cell to maintain a high osmotic pressure, allowing it to retain water. When solid stress on the cell’s surface increases, its internal hydrostatic pressure also increases (Supplementary Fig. 1b) and water is squeezed from the cell (Fig. 2c); recall that water flow is partly driven hydrostatic pressure differences (Eq. 4). The loss of water relieves tension in the cell’s membrane, thereby reducing the permeability of its MS channels (via Eq. 5). However, as ion pumping remains active, there is a continuous intake of ions (Supplementary Fig. 1f). Overall, the cell loses fewer ions, but the rate of ions entering the cell remains relatively constant and therefore there is a net increase in its ion concentration. This increases the loaded cell’s osmotic and hydrostatic pressure (Supplementary Fig. 1b, c) relative to its connected neighbor, and the differences generate a flow of ions through their GJs from the loaded to the unloaded cell (Fig. 2d). The resulting increase in the unloaded cell’s ion concentration causes it to absorb additional water from the external media and swell (Fig. 2b). Although its MS channels are now more permeable due to increased membrane tension, the constant flow of ions from the loaded cell maintains the neighbors swollen state. Thus, compression of a cell increases its osmotic pressure relative to its connected neighbor due to closing of MS channels. This drives a flow of ions across GJs into the unloaded cell, causing it to absorb water and swell.

To further analyze the role of cell–cell transport in volume regulation, we inhibit GJs by reducing their ion and water permeability (via \(\omega _{\mathrm{g}}\) and \(L_{{\mathrm{p}},{\mathrm{g}}}\), respectively) to zero. The loaded cell shrinks (Fig. 2e), but it loses less water than it would under control conditions. This reduction in volume again relieves membrane tension, reducing the permeability of MS channels and increasing the cell’s overall ion concentration. However, as GJs are blocked there is no loss of ions to the neighboring cell (Fig. 2g) and thus the loaded cell sustains its high osmotic pressure (Supplementary Fig. 2c). This allows it to retain more water (and maintain a high hydrostatic pressure) to oppose the applied compressive stress. Additionally, there is no swelling predicted in the neighboring cell (Fig. 2e), owing to the lack of cell–cell ion exchange. Clearly, ion transport across cellular GJs plays an important role in cellular volume regulation, with an increasing GJ permeability driving larger differences in cell volume (Supplementary Fig. 3b). Although we assume that cells retain an approximately spherical shape, our simulations also suggest that similar trends emerge in connected elongated cells (Supplementary Fig. 4).

In this analysis, we have implicitly assumed that the external osmotic and hydrostatic pressures are spatially uniform (i.e. act equally on both connected cells). However, this may not always hold true, especially in large organoids where media perfusion can be impaired. We therefore explore the influence of spatial variance of osmotic pressure and hydrostatic pressure in Supplementary Notes 9 and 10, respectively, in the context of mechano-electro-osmotic flow. Our analyses reveal that hydrostatic pressure gradients have an impact similar to differences in applied solid stress of equivalent magnitudes (Supplementary Fig. 8c); briefly, increased external hydrostatic pressure in a loaded cell drives ions across GJs into a connected neighbor, which consequentially swells. Therefore, consideration of solid growth stress can also implicitly describe the influence of variations in external fluid pressure. Conversely, equivalent differences in osmotic pressure have a markedly lower influence on cellular volumes due to the ability of cells to adapt in response to osmotic shocks (Supplementary Fig. 7). In the next section, we proceed to generalize our analysis of volume changes in a two-cell system to a multicellular cluster.

Intercellular ion transport drives spatial variations in cell volume within proliferating solid tumors

Multicellular spheroids are an increasingly promising experimental model for cancer development, that aim to bridge the gap between in vitro and in vivo conditions¹¹. In such 3D environments, however, it remains unclear how individual cells regulate their volumes and coordinate to advance tumor progression and matrix invasion. Therefore, we next consider how our mathematical framework can be extended to predict fluid and ion exchange within connected cells in a tumor organoid (as driven by differences in solid stress, hydrostatic pressure, and osmotic pressure). While our current model can readily be adapted to simulate a series of discrete connected cells, more physical insights can be gained from a continuum formulation that describes cellular behavior within a spherical organoid (Fig. 3a). Continuity requires that for any cell within a cluster, the change in its number of ions must equal the amount gained and lost through GJs to neighbors and through the cell membrane. First, considering cell–cell fluid exchange, recall that the volume flux across GJs between two connected cells depends on their hydrostatic pressure difference, with \(J_{{\mathrm{v}},{\mathrm{g}},i} = - L_{{\mathrm{p}},{\mathrm{g}}}\left( {P_i - P_{i + 1}} \right)\). The gradient of hydrostatic pressure between these cells may be written as \(\nabla \left( {{\Delta} P} \right) = \left( {{\Delta} P_i - {\Delta} P_{i + 1}} \right)/r_0\). For a cell within a longer series, this can be extended to develop an expression for the Laplacian of the hydrostatic pressure, whereby:

a Schematic of model extension to 3D spherical organoid. b Applied solid growth stress \(\sigma _{\mathrm{g}}(r)\) is highest at the core and spatially non-uniform. c Predicted spatial cell volume. d Predicted and experimental (day 5) spatial nuclear volume in the organoid. Predicted spatial e osmotic pressure \({\Delta} {\Pi}\) and f channel permeability \(f = \omega _{{\mathrm{ms}}}\left( \sigma \right) + \omega _{\mathrm{l}}\). g Cross-section images of epithelial cancer organoid developed from MCF10A cells at day 5 of growth with GFP-NLS-labeled cell nuclei. Scale bar 50 μm. h Nuclear volume of cells at organoid core and periphery at day 5 of growth (n = 5 multi-cellular clusters over three independent experiments). The boxes represent the interquartile range between the first and third quartiles, whereas the whiskers represent the 95% and 5% values, the squares represent the median, and the shaded region bounds indicate the maxima and minima. A two-tailed Student’s t-test was used when comparing the difference between two groups *P < 0.001.

In a spherical coordinate system, assuming circumferential symmetry, this Laplacian can be rephrased as \(r_0^2\nabla ^2\left( {{\Delta} P} \right) = \frac{{r_0^2}}{{r^2}}\frac{\partial }{{\partial r}}\left( {r^2\frac{{\partial \left( {{\Delta} P(r)} \right)}}{{\partial r}}} \right)\), where r is the radial position in a spherical organoid. Further, the volume flux across GJs may also be extended to describe a cell in-series and connected on both sides, such that \(J_{{\mathrm{v}},{\mathrm{g}},i} = - L_{{\mathrm{p}},{\mathrm{g}}}\left( {\left( {{\Delta} P_i - {\Delta} P_{i - 1}} \right) + \left( {{\Delta} P_i - {\Delta} P_{i + 1}} \right)} \right)\). Altogether, we can then enforce continuity to formulate a continuum expression that describes cellular volume at position r within a multicellular spheroid, such that

where the first term on the right describes the volume change driven by fluid flow through GJs in a cell at position \(r\), and the second term accounts for cellular exchange of fluid with the surrounding media. Similarly, the number of ions entering or leaving a cell may be expressed by

where the first term on the right describes the ions gained and lost through GJs in a cell at position r (following from Eq. 6), and the second term accounts for cellular exchange of ions with the surrounding media via pumps and channels. We solve our system of equations at steady state (i.e. \(\frac{{\partial N\left( r \right)}}{{\partial t}} = \frac{{\partial V_{\mathrm{c}}\left( r \right)}}{{\partial t}} = 0\)) using multi-physics software COMSOL to simulate local cell behavior within a spherical organoid of radius \(r_{\max }\) (see ‘Methods for more details). As the GJ flux vanishes at the cluster boundary, a zero-flux condition is enforced on the spheroid surface.

Proliferation of cells within a growing cluster generates solid compressive stresses, additionally compounded by matrix stretch and cell confinement. Interestingly, it has been shown that such local compressive stresses are spatially non-uniform across the cancerous structure^46,47, frequently highest at the cluster core and lowest at the periphery, with maximum stress \(\sigma _{\mathrm{g}}^{\max }\). We consider this compressive stress \(\sigma _{\mathrm{g}}(r)\) to act uniformly on the surface of a cell located at position r in the spheroid. To estimate the distribution of compressive solid stresses in our multicellular clusters, we simulate spheroid growth in a hydrogel system (Supplementary Note 12). Simulations suggest that the solid stress varies from ~550 Pa at the core to ~200 Pa at the periphery, which we apply in our analysis (Fig. 3b). Our model predicts that cell volume is lowest at the core (\(V_{\mathrm{c}}\left( {r = 0} \right) = 750\;\upmu {\mathrm{m}}^3\)) and increases radially (\(V_{\mathrm{c}}\left( {r = r_{\max }} \right) = 2000\;\upmu {\mathrm{m}}^3\)), as shown in the organoid contour plot in Fig. 3c. We have shown that the ratio of nuclear to cell volume remains constant during volume changes¹², maintaining a value of \(V_{\mathrm{n}}:V_{\mathrm{c}} \cong 0.14\). Applying this ratio, we can predict nuclear volumes across the organoid (Fig. 3d), and show strong agreement with our recent experiments¹² as discussed in more detail in the next section. At the core, where there is the most significant cell shrinkage, water loss reduces the membrane tension and thus MS channel permeability is impaired (Fig. 3f). As a result, the osmotic pressure of internal cells increases (Fig. 3e). Radially, the solid growth stress reduces, and ion channels become increasingly permeable provided the membrane stress exceeds the critical value \(\sigma _{\mathrm{c}}\) (Eq. 5). Within the whole organoid, this generates an intercellular ion concentration gradient that propagates radially (Fig. 3e), driving an ion flow that increases the osmotic pressure in peripheral cells and causes them to absorb water. The loss of ions from core cells impedes their ability to retain water, and thus they are highly compressed during loading. As such, at steady state there is a balance attained between the solid growth stress (that drives a water efflux) and cellular ion concentrations (that drive a water influx). During an early stage of organoid growth when solid stress is low and approximately uniform, there is no predicted spatial variation in cell volume (Supplementary Fig. 10).

Further, in recent work¹², we obtained grade2 ER+ invasive ductal carcinoma breast cancer samples from a human patient, which were fixed, sectioned, and stained for imaging with confocal microscopy (Supplementary Fig. 11a). Within the tumor mass, spheroidal acinar-like clusters of cells surrounded by basement membrane were identified, which share characteristics with the 3D experimental model data reported in this study. In these acinar-like structures, nuclear volumes were observed to increase from core to periphery, consistent with our simulations (Supplementary Fig. 11b). In summary, our model suggests that ion flow through GJs, as generated by differences in cellular ion concentrations, is a significant driving factor in cell swelling and shrinkage within multicellular organoids. In the next section, we discuss additional recent experiments to validate model predictions.

Experimental evidence validates model predictions that GJs mediate cell swelling in multicellular spheroids

In recent work, we experimentally uncovered that locality within a breast cancer organoid governs cell volume and stiffness¹². We seeded single MCF10A human breast epithelial cells into 3D hydrogels composed of 4 mg/ml Matrigel and 5 mg/ml alginate (see Methods for details), such that the gels had a shear modulus of approximately 300 Pa to reflect the environment of in vivo breast carcinoma⁴⁸. Initially, an isolated cell proliferated to form a spherical cluster (day 3), continuing to grow into a larger spheroid (day 5) with cells present both in the core and at the periphery (Fig. 3g). Further growth led to invasive branches extending into the surrounding matrix. Cells were transfected with a green fluorescent protein (GFP) tagged nuclear-localization signal (NLS), which enabled measurement of nuclear volume using 3D confocal microscopy (Fig. 3g). Importantly, we identified that the ratio of nucleus to cell volume remained constant over a wide range of organoid sizes and cell positions¹², in agreement with previous findings^49,50, which allowed us to measure nuclear volume in lieu of cell volume. At an early stage of growth, all cells had similar nuclear volumes (Supplementary Fig. 10e). However, as the organoid further developed, nuclear volume began to correlate strongly with cell position within the cluster. Cells toward the core (inner 40% of organoid radius) were significantly smaller than those in peripheral regions (Fig. 3h), with a volume range in strong agreement with our model predictions (Fig. 3d). Interestingly, we also demonstrated that a reduction in volume was highly correlated with an increase in cell stiffness (measured using optical tweezers active microrheology⁵¹), most likely due to increased molecular crowding^49,52. These experimental results both validate our model findings and highlight the importance of understanding the mechanisms of volume change within multicellular spheroids.

As our model suggests that GJs are a critical mediator of spatial volume variation within breast cancer organoids, we next explore how cell behavior changes when GJs are blocked. The spatial variation in cell volume is shown to decrease significantly (Fig. 4a), ranging from ~1500 μm³ at the core to ~1550 μm³ at the periphery. As per the control case (active GJs), compressive stress reduces cell volume throughout the spheroid. However, when the MS channels of inner cells close (Fig. 4d), there is no loss of ions through GJs to relieve the cells’ high ion concentrations. Thus, relative to the control case, cellular osmotic and hydrostatic pressure are significantly higher at the organoid core (Fig. 4e). Further, peripheral cells have a low osmotic pressure because they do not gain ions from their neighbors; therefore they do not absorb water and swell. In fact, they retain a volume similar to that during an early stage of organoid growth (Supplementary Fig. 10e). In our experiments, we inhibited GJs by adding 500 μM carbenoxolone to the organoid-matrix system on day 3 (before a volume gradient was present)¹². In agreement with our simulations, on day 5 we did not observe significant spatial differences in nuclear volume (Fig. 4a, c). This further supports our model findings that ion diffusion driven by an intercellular osmotic pressure gradient is the critical factor in cell swelling and shrinkage within the organoid.

Finally, as cell volume clearly additionally depends on the solid stress gradient, we examined the influence of reducing the solid stress acting on the cell cluster. Our model predicts that a reduction in the maximum (core) solid stress (\(\sigma _{\mathrm{g}}^{\max } = 400\;{\mathrm{Pa}}\)) also reduces the spatial variation in cell volume (Fig. 4b). The reduction in stress allows more cells to sustain their MS channel permeability (Fig. 4d), permitting a loss of ions to the interstitium and thereby lowering their osmotic pressure. This reduces intercellular ion diffusion and thus lowers cell swelling in peripheral regions of the cluster. We performed additional experiments that can test these predictions; organoids were cultured in a collagen (3.5 mg/ml)/Matrigel (0.5 mg/ml) matrix to achieve a day-5 nucleus volume distribution similar to the Matrigel/alginate system (Supplementary Fig. 12). We then reduced solid stress on day 5 by degrading collagen fibers with collagenase. After 6 h, we observed a significant increase in nuclear volume at the core and a reduction at the periphery (Fig. 4b, c); the volume gradient became weaker, in agreement with model findings. In summary, the spatial variation in cell volume within the breast cancer organoid may be understood to depend on intercellular ion diffusion in response to an osmotic pressure gradient, as driven by non-uniform external solid stress.