Intercellular water exchanges trigger soliton-like waves in multicellular systems

Abstract

Oscillations and waves are ubiquitous in living cellular systems. Generations of these spatiotemporal patterns are generally attributed to some mechanochemical feedbacks. Here, we treat cells as open systems, i.e., water and ions can pass through the cell membrane passively or actively, and reveal a new origin of wave generation. We show that osmotic shocks above a shock threshold will trigger self-sustained cell oscillations and result in long-range waves propagating without decrement, a phenomenon that is analogous to the excitable medium. The traveling wave propagates along the intercellular osmotic pressure gradient, and its wave speed scales with the magnitude of intercellular water flows. Furthermore, we also find that the traveling wave exhibits several hallmarks of solitary waves. Together, our findings predict a new mechanism of wave generation in living multicellular systems. The ubiquity of intercellular water exchanges implies that this mechanism may be relevant to a broad class of systems.

Significance

Mechanical responses of living cells are quite different from non-living materials. The oscillation patterns in living cellular systems are highly organized in space and time. Understanding the mechanisms responsible for these spatiotemporal pattern formations remains one of the outstanding challenges at the interface between physics and biology. In this work, we treat living multicellular tissues as non-equilibrium open systems and show that intercellular water flows can trigger self-sustained cell oscillations and soliton-like wave patterns in multicellular systems. Hence, our work indicates a new mechanism of tissue oscillations.

Introduction

Oscillations in living cellular systems are highly orchestrated in space and time. These spatiotemporal patterns are essential for cells to robustly cooperate toward large-scale collective behaviors. For instance, shape oscillations can drive the dorsal closure of amnioserosa tissue, a tissue-modeling process in early-developing Drosophila embryos (1,2). The oscillatory dynamics of myosin allows Drosophila embryos to instruct endoderm invaginations (3). Oscillations of lumenal pressure are harnessed to control embryo size and cell fate (4). Furthermore, in the presence of spatial coupling, local oscillation naturally triggers waves, such as propagating action potentials (5,6), calcium waves (7, 8, 9), actin waves (10,11), and mechanical-deformation waves (12, 13, 14). These traveling waves are important dynamic modes for reliable communication over large distances.

Feedback mechanisms are a common origin of oscillations and waves in multicellular systems. (15,16). For example, the mechanochemical feedback between cell deformation and active myosin contraction induces spontaneous oscillations of the cell cytoskeleton (17) and Drosophila amnioserosa (18). Likewise, this feedback causes a propagating wave in the actin cortex (11), cell cytoplasm (19), and cell monolayer (20). These studies treat cells as close systems and usually ignore the changes of cell volume under mechanical deformations. However, living cells can actively exchange water and ions with external cellular microenvironments, indicating that cells are intrinsically open systems far from thermodynamic equilibrium. Thus, the responses of cells to mechanical forces are quite different from non-living materials (21, 22, 23).

As open systems, cells dramatically change their volume during active shape oscillations. For example, cell volume fluctuates up to 10% during bleb expansions and retractions (24). There are large fluctuations in cell area over hours during dorsal closure (2), and cell volume decreases up to 50% in this process (25). Cell volume oscillates with a period of 4 h and an amplitude of 20% in the Madin Darby canine kidney monolayer (26). Over these long time scales, cells behave as active materials (20,27), and the fluctuation-dissipation theorem breaks down (28,29). Moreover, there are extensive exchanges of water and ions between adjacent cells in living multicellular systems (30, 31, 32, 33). Together, these findings pose a pivotal question: can active volume changes, induced by intercellular water and ion exchanges, trigger rhythmic oscillations and waves in multicellular systems?

Materials and methods

Model

We address this fundamental problem by studying the wave generation and propagation of multicellular systems under osmotic shocks. Similar to the notochord of Ciona intestinalis with 40 cells in length and a single cell in diameter (34,35), we here focus on one-dimensional multicellular chains confined in a channel (Fig. 1). A similar one-dimensional setup has also been used as a minimal system to study the mechanical communication between cells both theoretically and experimentally (36, 37, 38, 39). In multicellular systems, there are extensive exchanges of water and ions between adjacent cells (30, 31, 32, 33). If we assume there is no water and ion transport across the channel walls, the cell volume or length is completely determined by intercellular exchanges of water and ions (Fig. 1). Thus, the dynamics of cell lengths and ion numbers due to these interfacial fluxes are given as

$$\frac{d{L}_i}{dt}=J_{water,i}-J_{water,i+1},\text{and}$$ (1)

$$\frac{d{n}_i}{dt}=WH\left(J_{ion,i}-J_{ion,i+1}\right),$$ (2)

where $L_i$ and $n_i$ are the length and ion number of the ith cell, respectively. W and H are the width and height of the channel, respectively. $J_{water,i}$ ($J_{ion,i}$) and $J_{water,i+1}$ ($J_{ion,i+1}$) denote the interfacial water flux (ion flux) on the left and right boundaries, respectively, of the ith cell (Fig. 1).

Figure 1.

Schematic of multicellular chain confined in a channel. Cells completely occupy the channel cross section, thus the cell length $L_i$, channel width W, and channel height H characterize cell shapes. There are interfacial water fluxes $J_{water,i}$ and ion fluxes $J_{ion,i}$ at cell-cell interfaces. The positive signs of these fluxes are defined as the right direction. ${\text{Π}}_L$ and ${\text{Π}}_R$ denote environmental osmotic pressures at the left and right sides of the channel, respectively. To see this figure in color, go online.

The interfacial water flux, driven by the osmotic- and hydrostatic-pressure differences between adjacent cells, is given as

$$J_{water,i}=\alpha \left[\left({\text{Π}}_i{\text{-Π}}_{i-1}\right)-\left(P_i-P_{i-1}\right)\right],$$ (3)

where α is a rate constant and ${\text{Π}}_i$ and $P_i$ are the osmotic pressure and hydrostatic pressure, respectively, of the ith cell. Note that ${\text{Π}}_0={\text{Π}}_L$, ${\text{Π}}_{n+1}={\text{Π}}_R$, and $P_0=P_{n+1}=P_{out}$, where ${\text{Π}}_L$ and ${\text{Π}}_R$ are the environmental osmotic pressures at the left and right sides of the microfluidic channel, respectively, and $P_{out}$ is the environmental hydrostatic pressure. For a dilute solution, the osmotic pressure is given by the Van’t Hoff equation as ${\text{Π}}_i=n_{i}RT/\left(L_{i}WH\right)$.

The interfacial ion flux $J_{ion,i}$ includes both the contribution from the active and the passive transports and is given as

$$J_{ion,i}=\left(J_{active,i}+J_{MS,i}\right),$$ (4)

where $J_{active,i}$ and $J_{MS,i}$ denote the active ion fluxes through active pumps and the passive ion fluxes through mechanosensitive (MS) channels, respectively, at the ith interface.

Active pumps are required to maintain the osmotic difference between the intracellular cytoplasm and the external environment since they can use energy to actively transport ions against the osmotic-pressure difference (40, 41, 42). Thus, if there is an osmotic-pressure difference between two adjacent cells, there should also exist active ion transport at the cell-cell interface to transport ions against the osmotic-pressure gradient to break the symmetry. We denote these active ion fluxes as $J_{active,i}$. Given that the positive sign of interfacial flux is defined as the right direction, $J_{active,i}$ is positive when ${\text{Π}}_i>{\text{Π}}_{i-1}$, and $J_{active,i}$ is negative when ${\text{Π}}_i<{\text{Π}}_{i-1}$. Therefore, the active pumping flux at the ith cell-cell interface can be written as (43,44)

$$J_{active,i}=\text{sign}\left({\text{Π}}_i{\text{-Π}}_{i-1}\right)\gamma \left[\text{Δ}{\text{Π}}_c\text{-}\left({\text{Π}}_i{\text{-Π}}_{i-1}\right)\right]\left(i=1\sim n+1\right),$$ (5)

where γ is a rate constant and $\text{Δ}{\text{Π}}_c$ is a critical osmotic-pressure difference, above which the energy input from ATP hydrolysis is insufficient to transport ions against the osmotic-pressure gradient.

The change of free energy when ions are actively pumped from cell i - 1 to cell i is $\text{Δ}G=RT\text{ln}\left(c_i/c_{i-1}\right)\text{-Δ}{G}_a=RT\text{ln}\left({\text{Π}}_i/{\text{Π}}_{i-1}\right)\text{-Δ}{G}_a$ (43,45), where the first term denotes the chemical potential and $\text{Δ}{G}_a$ denotes the free energy of ATP hydrolysis. The active ion pumping is energetically favorable only when $\text{Δ}G<0$. However, if the osmotic-pressure difference becomes extremely high (${\text{Π}}_i/{\text{Π}}_{i-1}\gg 1$), the chemical potential of ions will exceed the energy input from ATP hydrolysis, then $\text{Δ}G>0$, and the pumping flux reverses direction. We can obtain the critical osmotic-pressure difference, $\text{Δ}{\text{Π}}_c$, at which $\text{Δ}G=0$, that is $\text{Δ}{\text{Π}}_c={\text{Π}}_{i-1}\left[\text{exp}\left(\text{Δ}{G}_a/RT\right)\text{-}1\right].$ The free energy of ATP hydrolysis under standard cellular conditions is $\text{Δ}{G}_a\approx$ 30 kJ/mol (45,46), and the cellular osmotic pressure is on the order of $0.1\sim 1$ MPa (47), thus $\text{Δ}{\text{Π}}_c$ is on the order of 10–100 GPa, and we set $\text{Δ}{\text{Π}}_c=30$ Gpa.

MS channels would open when the surface tension of cells is big enough (40). After opening, the ion flux through the MS channels is driven by the osmotic-pressure difference across the ith interface. Thus, $J_{MS,i}$ can be modeled as

$$J_{MS,i}=-\beta P_{open}\left(\sigma \right)\left({\text{Π}}_i{\text{-Π}}_{i-1}\right)\left(i=1\sim n+1\right),$$ (6)

where β is a rate constant and $P_{open}$ is the opening probability of the MS channels for a given cortical stress σ.

The opening probability of MS channels follows a Boltzmann function of cortical stress σ (48). Thus, $P_{open}$ can be modeled as

$$P_{open}=\frac{1}{1+\text{exp}\left[a\left({\sigma }_m-\sigma \right)\right]},$$ (7)

where a is a constant, ${\sigma }_m$ is the midpoint stress when $P_{open}=0.5$, and σ is the cortical stress that governs the opening probability of the MS channel (see supporting material for details).

The force balances of cells read

$$\lambda \frac{d{L}_i}{dt}=f_{i+1}-2f_i+f_{i-1},$$ (8)

where λ is the frictional coefficient between cells and the microfluidic channel and $f_i$ is the equivalent force applied by the ith cell on the cell-cell interface. Both the cortical stress, ${\sigma }_i$, and the hydrostatic pressure, $P_i$, contribute to the equivalent force. Therefore, $f_i=2{\sigma }_{i}h\left(W+H\right)-P_{i}WH,$ where h is the thickness of the cortical layer (see supporting material for details). Notably, here we assume a cell-length-independent friction force. The influences of cell-length-dependent friction force are discussed later in “Cell-length dependent friction force” in the supporting material. These discussions demonstrate that the qualitative results do not depend on the assumptions of friction forces.

Cells are very spindly due to the confinement of the microchannel. Thus, we can define the cell strain by the change in cell length, and the constitutive law of each cell is given as

$${\sigma }_i=E\left(\frac{L_i}{L_{0,i}}-1\right)+{\sigma }_a,$$ (9)

where E is the elastic modulus of cortical layer, $L_{0,i}$ is the reference length of the ith cell, and ${\sigma }_a$ is the active stress due to myosin contraction. In previous models, the active stress is assumed to couple with cell deformation (17,19,20,44). Here, we assume a constant active contractile stress to study the wave generation independent of the mechanochemical feedback between active contraction and cell deformation.

Results

Hypotonic shocks with various shock amplitudes result in three distinct wave patterns

We first investigate the oscillatory dynamics of the multicellular chain after applying a hypotonic shock (osmotic pressure changes suddenly) to the left side of channel. Here, we assume that, initially, environmental osmotic pressures at the left and right sides are equal, i.e., ${\text{Π}}_L={\text{Π}}_R$ (Fig. 1), and that cells have reached their steady states before the hypotonic shock. We find that the probability density function of steady cell length is independent of the values of environmental osmotic pressures as long as ${\text{Π}}_L={\text{Π}}_R$ (Fig. S3). This finding is reminiscent of cell volume homeostasis, i.e., the steady cell volume is independent of environmental osmotic pressure (40,41,43).

Then, we decrease ${\text{Π}}_L$ to ${\text{Π}}_L\text{-}A,$ where A is the amplitude of the hypotonic shock. For small A, there is no traveling wave but a perturbation in cell lengths with decaying oscillation amplitude from the first to the last cell (Fig. 2 a). The changes in cell length after the osmotic shock are controlled by the interfacial water flows, which are directly related to the osmotic-pressure differences between cells. All cells in the channel show a drop in osmotic pressure right after the hypotonic shock, but the magnitude of this osmotic-pressure drop is smaller when the cell is farther away from the shock (Fig. 2 d). These differences in osmotic pressures cause the attenuation of net water inflows from the first to the last cell (Fig. S4 b); that is why the oscillation amplitude of cell-length decay as the cell becomes farther from the shock (Fig. 2 a).

Figure 2.

There are three distinct wave patterns after a hypotonic shock. (a–c) Spatiotemporal patterns of cell lengths $L_i$, normalized by steady lengths $L_{s,i}$ before the hypotonic shock, for various hypotonic shock amplitudes A. (a) A = 1 × 10⁴ Pa, (b) A = 4 × 10⁴ Pa, and (c) A = 1 × 10⁵ Pa. The number of cells is n = 20. Environmental osmotic pressures before shock are ${\text{Π}}_L={\text{Π}}_R=5\times {10}^5\text{Pa},$ and the hypotonic shock is applied to the left side of the channel at $t=2000$ s (indicated by red arrows in a–c). Other parameters are shown in Table S1 of the supporting material. (d–f) Time series of cell osmotic pressure (normalized by steady values) corresponding to (a)–(c); the color codes represent the indexes of cells. To see this figure in color, go online.

After the perturbation, all cells reach their new steady lengths. However, the new steady lengths of most cells are smaller than their initial steady lengths before the hypotonic shock ($L_i/L_{s,i}<1$), which is consistent with recent experimental observations that cells shrink in response to hypotonic shocks in narrow channels (49). Similar results have also been observed experimentally for unrestricted cells, where the new steady volume after hypotonic shocks is smaller than that of the isotonic value, termed regulatory volume decrease “overshoot” (50, 51, 52). Thus, these experimental observations indicate that the result of smaller new steady cell lengths after hypotonic shocks in our simulations is reasonable.

Strikingly, if A increases to $4\times {10}^4$ Pa, a traveling wave appears. In this case, there is a second perturbation in cellular osmotic pressures (Fig. 2 e). Specifically, the osmotic pressures increase in sequence from the last to the first cell, accompanied by oscillations in interfacial water flows and ion flows (Fig. S5 b and c). Thus, cells show a pulse in length one by one from the last to the first cell, resulting in a solition-like left-traveling wave (Fig. 2 b). After the hypotonic shock, there is also a left-traveling wave for the velocity of cell centroids (Fig. S6). Cells move one by one toward the right side of the channel from the last cell (Fig. S6).

Unexpectedly, for sufficiently large A, there are multiple traveling waves after a single hypotonic shock (Fig. 2 c). In this case, the lengths of several cells at the left margin of the channel still oscillate after each left-traveling wave, propagating to the left side of the channel (Fig. 3 a–c). These oscillations give rise to multiple perturbations in cellular osmotic pressures (Fig. 2 f), which then lead to multiple left-traveling waves. This is reminiscent of the damped oscillation, which also shows multiple oscillations for a big shock amplitude. However, in contrast to non-living -systems, there are continuing energy supplies from ATP hydrolysis in living multicellular systems, thus a single pressure shock could generate an infinite number of waves. It should be also noted that these multiple waves are non-periodic since the time intervals between two adjacent waves are not exactly the same (Fig. 3 d). However, the waveform, wave amplitude, and wave speed of individual waves are almost identical (Fig. 3 a–c and e–f).

Figure 3.

The multiple waves are non-periodic. (a–c) Time series of representative waves with different wave indices. (d–f) Quantifications of the time interval, normalized wave amplitude $A_w$ (normalized by the mean amplitude), and normalized wave speed $v_w$ (normalized by the mean speed) of the multiple waves. To see this figure in color, go online.

It should be noted that previous studies show that active ion transport across the membrane will not be activated by a gradual change of extracellular osmolarity (53). Thus, cells respond differently to a suddenly or gradually changed osmotic pressure. Analogously, here we find that a gradually changed osmotic pressure cannot trigger traveling waves even if the changing amplitude of the osmotic pressure is big enough (Fig. S7).

Together, these results show that there is a wave propagating without decrement above a certain threshold amplitude of hypotonic shocks, which is similar to the characteristic response properties of excitable medium (54,55). An excitable medium is a non-linear dynamic system, each element of which can be activated. This medium returns to its steady state after small perturbations. For sufficiently strong perturbations, however, the excitable medium diverges from its steady state. In the presence of spatial coupling (here, the coupling results from intercellular exchanges of water and ion), a local excitation can trigger excitations at neighboring locations, resulting in undamped waves propagating at a constant speed.

Phase diagrams of wave patterns

Besides A, the number of cells (the length of cell chain) can also affect the number of traveling waves. As shown in the phase diagram of wave patterns (Fig. 4 a), the critical shock amplitudes across different wave patterns increase with the number of cells. This result indicates that it will need a bigger shock amplitude to trigger a traveling wave when there are more cells in the channel.

Figure 4.

Phase diagram of wave patterns. (a) The phase diagram in the space of osmotic shock amplitude A and cell number. (b) The phase diagram in the space of shock amplitude A and ion passive transport rate β, when γ = 10^-17 mol × m^-2 s^-1 Pa^-1. (c) The phase diagram in the space of shock amplitude A and ion active transport rate γ, when β = 10^-8 mol × m^-2 s^-1 Pa^-1. (d) The phase diagram in the space of β and γ, when $A=40$ kPa. Various symbols denote different wave patterns. To see this figure in color, go online.

We next discuss the impacts of rate constants β, γ, and the elastic modulus E on the generation of waves since these parameters are critical to the amplitudes of ion and water fluxes. The ion flux across MS channels is on the order of 10^–7–10^–6 mol × m^–2 s^–1 (56). Dividing the ion flux by P_open (0.1–1) and $\text{ΔΠ}$ ($\sim 100$ Pa), we find that the range of β is 10^–9 to 10^–7 mol × m^–2 s^–1 Pa^–1. We find that, for a given γ, both the critical shock amplitudes between the “zero” wave pattern and the “one” wave pattern and that between the one wave pattern and the “multiple” waves pattern become smaller as β increases (Fig. 4 b). Thus, a bigger β will promote the generation of waves.

The flux associated with active ion pumps has been measured to lie between 10^-7 and 10^-6 mol × m^-2 s^-1 (56). Dividing this flux by $\text{Δ}{\text{Π}}_c$ (10–100 Gpa) gives a range of γ from 10^-18 to 10^-16 mol × m^-2 s^-1 Pa^-1. We find that, for a given β, an increase in γ will suppress the generation of the one wave pattern under small shock amplitudes, whereas an increase in γ will facilitate the generation of multiple waves under big shock amplitudes (Fig. 4 c). Thus, for a small γ, the cell chain cannot generate multiple waves (light blue region in Fig. 4 c), whereas the cell chain cannot generate the one wave pattern for a big γ (light yellow region in Fig. 4 c). Only for a modest value of γ can the cell chain show three different wave patterns (light purple region in Fig. 4 c). Together, these findings indicate that a bigger ratio of β to γ is beneficial for the generation of waves (Fig. 4 d). It should be noted that the elastic modulus E hardly affects the wave patterns, but an increase in E will lead to a smaller wave amplitude (Fig. S8).

The wave propagates from the side with higher osmotic pressure to the side with lower osmotic pressure

An intriguing question is in which direction the wave should travel. We find that a hypotonic shock at the left side of the channel results in a left-traveling wave (Fig. 2), indicating that the wave does not always propagate from the disturbed side to the other side. Given that intercellular exchanges of water and ions are driven by intercellular osmotic-pressure differences, we speculate that the traveling wave propagates from the side with higher external osmotic pressure to the side with lower external osmotic pressure after the osmotic shock. Thus, we expect that a hypertonic shock at the left side of the channel will trigger a right-traveling wave. As shown in Fig. 5 a, this is indeed the case. Furthermore, in contrast to the right movements of cells after a hypotonic shock at the left side (Fig. S6 c), a hypertonic shock at the same side of the channel leads to left movements of cells (Fig. S9). This is consistent with the experimental observation that both a hypotonic shock at the leading edge and a hypertonic shock at the trailing edge of the cell reverse the direction of cell migration (49). Therefore, a hypotonic shock at the left side of the channel is equivalent to a hypertonic shock at the right side of the channel since they both result in smaller osmotic pressure at the left side of the channel and trigger a left-traveling wave (Fig. 5 b). Together, we conclude that after osmotic shocks, the traveling wave propagates toward the side with smaller environmental osmotic pressure.

Figure 5.

The traveling wave propagates from the side with higher osmotic pressure to the side with lower osmotic pressure. (a) Plots of normalized cell lengths after applying a hypertonic shock to the left side at $t=2000$ s with A = 4 × 10⁴ Pa (yellow arrow). (b) Schematic diagram showing why a hypertonic shock at the left side triggers a right-traveling wave. (c) Wave velocity as a function of osmotic pressure difference ${\text{Π}}_L{\text{-Π}}_R.$ Inset shows the definition of wave velocity $v_w$, where $L_t$ is the steady length of the multicellular chain and $\text{Δ}t$ is the time needed for the wave to propagate through the whole multicellular chain. The legends “left hypo” and “right hypo” indicate a hypotonic shock at the left and right sides of the channel, respectively, and “both hypo” indicates hypotonic shocks at both sides of the channel. To see this figure in color, go online.

To further verify this conclusion, we study the wave propagation after applying two osmotic shocks to two sides of the channel simultaneously. In this scenario, it is not the shock amplitude but the environmental osmotic-pressure difference after osmotic shocks that controls the wave generation and propagation (Fig. S10). There is no traveling wave if the osmotic-pressure difference is small, even if the shock amplitudes are very big (Fig. S10 a–d). Only when the osmotic-pressure difference is big enough will the traveling wave emerge (Fig. S10 e and f).

We further investigate how the wave velocity of the traveling wave $v_w$, defined by the ratio of the steady length of the multicellular chain $L_t$ to the time needed for the wave to propagate through the whole multicellular chain $\text{Δ}t$ (Fig. 5 c, inset), relates to the osmotic pressure differences after osmotic shocks. Here, a positive wave velocity denotes a right-traveling wave, while a negative wave velocity indicates a left-traveling wave. As shown in Fig. 5 c, the sign of $v_w$ agrees with the sign of ${\text{Π}}_L{\text{-Π}}_R$, which further confirms that the wave propagates from the side with higher external osmotic pressure to the side with lower external osmotic pressure.

The magnitude of $v_w$ increases with the osmotic-pressure difference after osmotic shocks (Fig. 5 c) since the interfacial water flows increase with the osmotic-pressure difference (Fig. S11 a and b). Furthermore, $v_w$ increases linearly with the rate constants of water and ion transportation (Fig. S11 e) because interfacial water and ion flows are linearly proportional to these rate constants (Fig. S11 f). These rate constants may vary widely across different tissues, which may lead to various wave speeds in different tissues. These results further confirm that the traveling wave is driven by intercellular exchanges of water and ions.

The traveling wave is a solition-like wave

As shown in Figs. 2 and 3, the traveling wave shows a single peak and maintains a constant waveform and speed, which are unique features of solitary waves (57,58). Moreover, we find that increasing osmotic-pressure difference also causes an increase in the wave amplitude $A_w$ (Fig. 6 a), and the wave velocity $v_w$ increases linearly with the wave amplitude $A_w$ (Fig. 6 b), which is also a unique characteristic of solitary waves. These findings thus indicate that the traveling wave we find here is a soliton-like wave.

Figure 6.

The traveling wave is a soliton-like wave. (a) The mean wave amplitude, $A_w$, scaled by the minimum amplitude, $A_w^{min}$, is proportional to the osmotic-pressure difference. (b) The wave velocity, $v_w$, scaled by the minimum velocity, $v_w^{min}$, increases linearly with the wave amplitude. (c and d) The traveling wave reverses its direction right after ${\text{Π}}_L{\text{-Π}}_R$ changes sign. The first left hypotonic shock is applied at $t=2000$ s with A = 4 × 10⁴ Pa (green arrows), and the second right hypotonic shock is applied at $t=3200$ (c) or $t=2700$ s (d) with A = 8 × 10⁴ Pa (red arrows). To see this figure in color, go online.

Another special feature of solitary waves is that two solitary waves retain their waveforms after colliding with each other. To test whether the traveling wave we find here also displays this feature, we apply two asynchronous hypotonic shocks on two sides of the channel to trigger two waves traveling in opposite directions (Fig. 6 c). If these traveling waves also possess the collision feature of solitary waves, there will be two opposite-traveling waves when the second hypotonic shock is applied during the propagation of the first wave. However, we find that there is only one traveling wave after applying the second osmotic perturbation (Fig. 6 d). The wave direction reverses right after the second osmotic perturbation since the sign of ${\text{Π}}_L{\text{-Π}}_R$ (i.e., the direction of water flows) reverses right after this osmotic shock (Fig. 6 d). Thus, these results indicate that the traveling wave we find here displays most of the characteristics of solitary wave, but it is not yet a solitary wave.

Discussion

The mechanochemical feedbacks between cellular deformations and active myosin contractions have long been considered as an origin of oscillation patterns in living tissues (17,18,20). However, cells can exchange ions and water with their adjacent cells (30, 31, 32, 33). How these exchanges impact the generation of waves is still poorly understood. Here, we show that intercellular exchanges of water and ions elicit a soliton-like wave pattern in multicellular systems. We expect that this novel spatiotemporal pattern may be of great general interest to physicists, biologists, and scientific enthusiasts, independent of its biological relevance. Previous experiments have also observed solition-like waves in various systems. For example, there are solition-like traveling waves of myosin II during Drosophila endoderm invagination (3). Meiosis in the starfish oocyte will trigger solition-like traveling waves of contraction (14). Apoptosis propagates in the cytoplasm of Xenopus laevis egg via solition-like traveling waves (59). During osteoblast regeneration, there are solition-like traveling waves of Erk activity (60).

Since we assume a constant active myosin contraction, the mechanism of wave generation we found here is independent of the mechanochemical feedback between cellular deformations and myosin contraction. Thus, our findings unveil a new mechanism for wave generations in multicellular systems and may have important implications for other processes, such as wound healing and tissue morphogenesis. Furthermore, the ubiquity of intercellular water flows indicates that this mechanism may be relevant for a broad class of systems.

We find that the traveling wave propagates along an intercellular osmotic-pressure gradient since intercellular water flows are driven by the osmotic-pressure gradient (Fig. 5), and stronger water flows result in bigger wave speeds. Furthermore, the traveling wave shows a single peak and maintains a constant waveform, and its amplitude is almost constant during its propagation. Thus, the traveling wave is a soliton-like wave and can provide a robust mode for cell-cell communication over large distances.

We also show that a single osmotic shock with big shock amplitude can generate an infinite number of waves (Figs. 2 c). In nonliving systems, a single shock can not give rise to infinite waves due to the dissipation of energy. However, in living cellular systems, cells can constantly to use the energy from ATP hydrolysis for energy supply. This major difference between living cells and non-living materials can account for why a single osmotic shock can lead to infinite waves in living multicellular systems.

In our model, we consider a one-dimensional chain of cells, embedded in a microfluidic channel, where there is no direct exchange of water and ions between the cells within the chain and the extracellular environment. In reality, cells in living tissues can exchange water and ions with extracellular space. These exchanges have been shown to play critical roles in the formation of enclosing lumens (61, 62, 63). Thus, it is possible to develop a more comprehensive model incorporating the exchange of water and ions with extracellular environment.

Author contributions

H.J. initiated and supervised the project. Y.Y. performed the simulations. Both authors analyzed the data and wrote the manuscript.

Editor: Jeremiah J Zartman.