Collective oscillations of coupled cell cycles

Abstract

Problems with networks of coupled oscillators arise in multiple contexts, commonly leading to the question about the dependence of network dynamics on network structure. Previous work has addressed this question in Drosophila oogenesis, in which stable cytoplasmic bridges connect the future oocyte to the supporting nurse cells that supply the oocyte with molecules and organelles needed for its development. To increase their biosynthetic capacity, nurse cells enter the endoreplication program, a special form of the cell cycle formed by the iterated repetition of growth and synthesis phases without mitosis. Recent studies have revealed that the oocyte orchestrates nurse cell endoreplication cycles, based on retrograde (oocyte to nurse cells) transport of a cell cycle inhibitor produced by the nurse cells and localized to the oocyte. Furthermore, the joint dynamics of endocycles has been proposed to depend on the intercellular connectivity within the oocyte-nurse cell cluster. We use a computational model to argue that this connectivity guides, but does not uniquely determine the collective dynamics and identify several oscillatory regimes, depending on the timescale of intercellular transport. Our results provide insights into collective dynamics of coupled cell cycles and motivate future quantitative studies of intercellular communication in the germline cell clusters.

Significance

Clusters of cells interconnected by stable cytoplasmic bridges provide an exciting opportunity for exploring the emergence of collective effects in small multicellular systems. Recent studies focused on cell cycles in the female germline cell cyst in Drosophila and suggested that the emerging oscillatory regime is dictated by the pattern of cluster connectivity. Our comprehensive numerical test of this proposal in four-cell networks reveals that collective oscillatory regimes are affected by the joint effects of cluster connectivity and intercellular transport.

Introduction

From insects to mammals, living systems begin their development as small clusters of cells connected through intercellular bridges (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14). Starting from just a single cell, these cysts are formed through multiple rounds of cell division without full cytokinesis (15). The resulting networks of connected cells can take on a variety of sizes and shapes, but they eventually converge to the same fate: the fertilizable egg (3,15). In many of these cysts, a single cell is selected as the oocyte, the precursor to the future egg, after the initial rounds of divisions. The remaining cells in the cyst take on a supporting role, continuing their cell cycles and providing the oocyte with proteins and other biomaterials necessary for further development (Fig. 1 A; (15,16)).

Figure 1.

(A) Schematic of the general process of oogenesis that occurs across a wide range of animal species. (B) Examples of germline clusters of various animals, highlighting the vast differences that can exist in size and shape. These networks can be long, linear chains (plumed worm (1)), branched trees (fruit fly, Argentine ant (3,16)), or inherently asymmetric (water bear (12)). (C) Schematic of interactions between the components of the model, both in the oocyte and within each supporting nurse cell. (D) Limit cycle for the interaction between X and Y in each nurse cell in the absence of Z, assuming constant values $\left(a,b,c,d,e\right)=\left(0.01,0.1,0.01,0.1,100\right)$.

Recent work has demonstrated that the oocyte in the developing Drosophila melanogaster cyst (Fig. 1 B), which arrests its cell cycle after selection, plays a direct role in coordinating the cell cycles of its supporting cells throughout the cluster (17). It accomplishes this by accumulating RNA synthesized in the supporting cells and producing a cell cycle inhibitor that diffuses throughout the cell network. This feedback yields coordinated groupwise oscillations in the supporting cells based on their distance from the oocyte. This proposed mechanism based on experimental observations was further supported through a modeling approach, in which each cell in the cyst was treated as a nonlinear oscillator and the intercellular bridges mediated coupling between adjacent nodes, allowing for intercellular transport.

This coupled oscillator model demonstrated that diffusion of an inhibitor throughout a network is sufficient to define groups of limit cycle oscillations based on distance from the central collection node. We propose that this mechanism may also operate in the developing cysts of other organisms. Although each Drosophila germline cyst undergoes the same pattern of initial divisions to create a maximally branched, 16-cell cyst, there is a wide diversity in the size and shape patterns of developing cysts of other species (see Fig. 1 B). The roles that network topology and size play in guiding individual cell cycles and groupwise behaviors of the developing cyst remain largely unknown. This is the question we address in this article.

Computational analysis of coupled oscillators has yielded insights into symmetries within natural and engineering systems, such as wave propagation on networks, chimera states, and the onset of synchronization (18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28). Here, we are interested in identifying the diversity of collective behaviors of different network structures in which the coupling between adjacent nodes allows for the diffusion of an inhibitor from a central node, which corresponds to what is observed in Drosophila oogenesis.

In this article, we first introduce a system of equations for the cell cycle within each supporting cell of the cyst, as well as the production and transport of an additional inhibitory component. Using this minimal model, we demonstrate that based solely on a single parameter comparing the timescales for degradation and diffusion of the inhibitory component, groupwise patterns can emerge within a maximally branched tree, as observed experimentally. From there, we focus on three archetypal network structures: tree, line, and star, analyzing each case to identify the various different behaviors that may be observed. Finally, we analyze the similarities and differences between each network type, paying special attention to the breaking of symmetry in patterns of solutions.

Materials and methods

Model description

During oogenesis, nurse cells (as supporting cells are called in Drosophila) enter a special type of cell cycle known as endoreplication, where each cell undergoes continuous growth and synthesis phases without intervening mitosis (29,30). This process, which is also known to be present in different cell types in other species, is governed by a class of regulatory proteins known as cyclins (17,31). These proteins activate a class of enzymes known as cyclin-dependent kinases (CDKs) by forming complexes that regulate other enzymes at different stages throughout the cell cycle. Importantly, the concentrations of these cyclins oscillate as the cell cycle progresses, through a combination of self-activation (in many cases) and through activating other molecules that inhibit cyclin activity (17,31,32). Therefore, we model the cell cycle as a coupled activator-inhibitor scheme to mimic these known biochemical interactions. This minimal model of the cell cycle is sufficient to create sustained oscillations in a single cell and lays the groundwork for understanding how connected oscillators may be affected by network coupling (33).

An additional RNA that encodes a CDK inhibitor within the developing Drosophila cell cluster is transcribed in nurse cells as a result of cyclin activity and transported directly to the oocyte (17,34). This RNA is translated only in the oocyte, and the protein product then distributes throughout the cell cluster via simple diffusion. The newly translated protein acts as a cell cycle inhibitor shared among the nurse cells, which is sufficient to create the groupwise coordination of cyclic behavior experimentally observed.

We now introduce a generalized model of the oocyte-nurse cell cluster, combining the effects of the intracellular activator-inhibitor and the intercellular diffusing inhibitor based on cell adjacencies. If the oocyte is taken to be cell 1 of the system, then for each nurse cell $i>1$, consider the interactions between an autocatalytic component $X_i$ and an inhibitor $Y_i$. Assume that production of $Y_i$ is induced by $X_i$, and that the oocyte produces an additional inhibitor $Z_1$ at a rate that is correlated with the total amount of $X_i$ that exists within the nurse cells. This mechanism is a simplification of previous work, in which component X produced a component W that directly localized to the oocyte to produce Z (17). By neglecting this component, we implicitly assume that the timescale for localization is much faster than the timescale of diffusion through the network, as well as reduce the number of equations needed to describe the temporal evolution of the network.

After accumulating in the oocyte, the inhibitor Z diffuses throughout the cell cluster where it may then further repress the production of $X_i$ within the nurse cells. To provide a concrete example for the identities of these components, in the Drosophila germline cyst X corresponds to the cell cycle protein Cyclin E, which forms a complex with CDK2, Y is a general placeholder for the various inhibitors of Cyclin E that help to induce oscillations, and Z represents Dacapo protein, a p27 homolog and specific Cyclin E/CDK2 complex inhibitor that also serves as an important cell cycle regulator (17,35,36).

After rescaling, the system of equations that describes the time evolution of these three components within the nurse cells and oocyte of a given cell cluster is as follows:

$$\frac{d{X}_{i>1}}{dt}=\left(\frac{a+X_i^2}{1+X_i^2}\right)\left(\frac{1}{1+Y_i}\right)\left(\frac{1}{1+Z_i}\right)-b{X}_i,$$ (1)

$$\frac{d{Y}_{i>1}}{dt}=c-d\frac{Y_i}{1+e{X}_i^2},$$ (2)

$$\frac{d{Z}_{i>1}}{dt}=-h{Z}_i+\frac{\gamma }{2}{\left[\underset{=}{\mathbf{B}}\mathbf{Z}\right]}_i,$$ (3)

$$\frac{d{Z}_1}{dt}=\alpha \sum _i{X}_i-h{Z}_1+\frac{\gamma }{2}{\left[\underset{=}{\mathbf{B}}\mathbf{Z}\right]}_1.$$ (4)

Here, $\underset{=}{\mathbf{B}}$ is the negative of the Laplacian matrix of the cell-cell connections within the cyst. For each connection between cell i and j, the value of entry $B_{ij}=B_{ji}=1$. In addition, if cell i has degree n, then we have $B_{ii}=-n$. This recurring factor of $1/2$ in Eqs. 3 and 4 comes from the symmetric transport between two adjacent cells and allows the intercellular transport of Z to be purely diffusive. Because $\mathbf{Z}$ is the vector of Z concentrations within all cells of the cyst, the product $\underset{=}{\mathbf{B}}\mathbf{Z}$ is itself a vector, where ${\left[\underset{=}{\mathbf{B}}\mathbf{Z}\right]}_i$ represents the ith component of this product.

The parameters $\left(a,b,c,d,e\right)$ are part of the activator-inhibitor oscillator model and set to $\left(0.01,0.1,0.01,0.1,100\right)$, sufficient to create a limit cycle between X and Y in the absence of Z, similar to the more complex model derived in (17). Although other parameter sets also permit limit cycles between X and Y, they produce the same qualitative behavior as these fixed choices. The degradation rate of Z in all cells is taken to be the same value, h, whereas transport between adjacent cells is given by rate γ. Z is synthesized only in cell 1, at a rate proportional to the sum of the concentrations of X in all nurse cells; for simplicity we assume that this proportionality constant $\alpha =h$.

Finally, as the timescale of oscillation is likely to be much longer than the timescale of diffusion of Z throughout the cyst, we introduce quasisteady state approximations for the concentrations of each $Z_i$ at all times; that is, we set the right-hand sides of Eqs. 3 and 4 to 0, and solve the resulting system of algebraic equations of each $Z_i$ in terms of $\sum X_i$ and $\delta =\gamma /h$. In this way, we reduce Eqs. 1, 2, 3, and 4 to a system of $2N$ equations for a cyst containing N nurse cells. For a fixed set of parameters describing the activator-inhibitor oscillator in each nurse cell, the dynamics of the cyst is determined by the single parameter $\delta =\gamma /h$, the ratio of the rate of transport of Z to its rate of degradation. Thus, we are left with the task of analyzing a dynamical system whose behavior relies entirely on the shape of the network and the value of a single parameter, δ.

Computational analysis

All systems of ODEs were implemented using the MATLAB (The MathWorks, Natick, MA) numerical time integration function ode45. Codes used to perform this analysis can be found at https://github.com/Shvartsman-Lab/NetworkTopology. All parameter continuation and limit cycle perturbation analysis was performed using MATCONT (37).

Results

Emergence of groups in the simplified model

To show that this model predicts the emergence of groups of synchronized cells within a maximally branched cyst, we analyzed the 16-cell network with the structure observed in Drosophila oogenesis. From the same set of initial conditions as previous numerical simulations, we found evidence of groupwise oscillations based on distance from the oocyte under the reduced model (17). As shown in Fig. 2, for $\delta =0.06$, the diffusing inhibitor has a propagating effect within the cell cluster, resulting in the blue cells firing nearly in unison, followed by the red, green, and yellow. This pattern of cell cycles emerges from randomized initial conditions, drawn from the uniform distribution $U\left(\mathrm{0,1}\right)+1.25$ for both the X and Y components within each nurse cell. This shows that our simplified model for inhibitory diffusion on a network can give rise to the groupwise oscillatory structure we are most interested in.

Figure 2.

Left schematic of the 16-cell cluster in Drosophila. Here, the gray cell of the network denotes the oocyte, whereas blue cells denote cells directly connected to the oocyte, red cells lie two connections away, green cells are 3, and, finally, yellow cells are 4. Right For a series of initial conditions for $\left(X_i,Y_i\right),i>1$ selected from the distribution $U\left(\mathrm{0,1}\right)+1.25$, using the systems of Eqs. 1, 2, 3, and 4 reveals oscillations within the network based on distance from the oocyte, similar to previous experimental observations for $\delta =0.06$ (17). To see this figure in color, go online.

Three types of four-cell networks

For the remainder of this work, we will focus on three distinct networks, each consisting of four nodes; that is, a single, nonreacting oocyte and three oscillating support cells. We refer to these three model geometries as the tree, the line, and the star (Fig. 3). We denote tree networks those with different numbers of nurse cells on the branches extending from the oocyte (Fig. 2), line networks those where the oocyte is at one end of a linear chain of cells, and star networks are those where the oocyte is the center node and all nurse cells are connected directly to it. These last two structures may be considered as two extremes of a spectrum that measures network symmetry: the line is maximally asymmetric with every cell uniquely identifiable based on its distance to the oocyte, whereas the star displays maximal symmetry with all cells being structurally equivalent. Below, we analyze how the collective oscillatory pattern of nurse cells are influenced by the joint effects of cluster connectivity and intercellular transport. The only differences between the systems of equations for each system is the term involving ${\left[\underset{=}{\mathbf{B}}\mathbf{Z}\right]}_i$, which matches the connectivity matrix for each case.

Figure 3.

Schematic (top) and the corresponding cell connectivity matrix $\underset{=}{\mathbf{B}}$ (bottom) for each of the three networks considered in this study. From left to right: the line, the tree, and the star. The gray circle represents the nonreacting oocyte, or cell 1 in our model. The nurse cells are numbered accordingly and distinguished by their colors in the subsequent figures. To see this figure in color, go online.

Tree network

The tree network can be thought of as a maximally branched structure, formed after a series of synchronous divisions. After applying the quasisteady state approximation for Z, we are left with the following set of six ODEs that describe the dynamics of X and Y within each supporting cell:

$$\frac{d{X}_2}{dt}=\left(\frac{a+X_2^2}{1+X_2^2}\right)\left(\frac{1}{1+Y_2}\right){\left(1+\frac{\delta {\left(2+\delta \right)}^2{\sum }_i{X}_i}{4\left({\delta }^3+5{\delta }^2+6\delta +2\right)}\right)}^{-1}-b{X}_2,$$ (5)

$$\frac{d{Y}_2}{dt}=c-d\frac{Y_2}{1+e{X}_2^2},$$ (6)

$$\frac{d{X}_3}{dt}=\left(\frac{a+X_3^2}{1+X_3^2}\right)\left(\frac{1}{1+Y_3}\right){\left(1+\frac{\delta \left(4+6\delta +{\delta }^2\right){\sum }_i{X}_i}{4\left({\delta }^3+5{\delta }^2+6\delta +2\right)}\right)}^{-1}-b{X}_3,$$ (7)

$$\frac{d{Y}_3}{dt}=c-d\frac{Y_3}{1+e{X}_3^2},$$ (8)

$$\frac{d{X}_4}{dt}=\left(\frac{a+X_4^2}{1+X_4^2}\right)\left(\frac{1}{1+Y_4}\right){\left(1+\frac{{\delta }^2\left(2+\delta \right){\sum }_i{X}_i}{4\left({\delta }^3+5{\delta }^2+6\delta +2\right)}\right)}^{-1}-b{X}_4,$$ (9)

$$\frac{d{Y}_4}{dt}=c-d\frac{Y_4}{1+e{X}_4^2}.$$ (10)

When $\delta =0$, it is clear that each oscillator behaves independently of the others. All oscillators trace the same limit cycle (column 1, Fig. 4). Therefore, for random and uniformly distributed initial conditions, the cells asynchronously enter the orbit and traverse it with an arbitrary phase shift.

Figure 4.

Phase plane, trajectories, and three-dimensional plot of $X_i$ for the tree network case. The black line in the first row of plots shows the unperturbed limit cycle. In all plots, blue corresponds to cell 2, red cell 3, and yellow cell 4 (see Fig. 3). Most notably, this system appears to undergo multiple regimes of quasiperiodic behavior before finally converging to a state where two cells are in sync and the third has a larger amplitude. To see this figure in color, go online.

Conversely, as $\delta \to \infty$, inhibitor concentration equalizes among all cells. As the speed of transport begins to dominate, it neutralizes the difference in distances between the oocyte and the nurse cells. Thus, we find that network shape does not affect the solution in this regime because of an identical degree of inhibition in each cell. For all initial conditions, the same limit cycle oscillation pattern is observed, in which one arbitrary cell oscillates about a high concentration of X, whereas the other two cycle synchronously around a lower concentration (column 4, Fig. 4). The two groups of cells are phase-locked with a half-period phase shift between them. We also find that distance from the oocyte does not dictate which cell will be the one to oscillate with a larger amplitude. This depends on initial conditions.

For intermediate values of δ, the behavior becomes less structured (column 2, Fig. 4). Although two of the cells are directly connected to the oocyte in the tree network, only the cell that is not connected to another nurse cell undergoes a regime with low amplitude and double frequency (column 3, Fig. 4). In this regime, the cells in the other branch of the tree oscillate in an antiphase manner. Additionally, as δ increases, each oscillator loses its limit cycle behavior, traversing instead a quasiperiodic or possibly chaotic trajectory. Further increasing δ results in the system converging to the $\delta \to \infty$ limit cycle behavior.

Line network

We now explore the oscillator dynamics in a cyst where the oocyte is placed at the terminal end of three nurse cells connected in a line. The line, although structurally different from the tree, seems to share many of its dynamical properties. Applying the same approximation for Z yields the following equations for X and Y:

$$\frac{d{X}_2}{dt}=\left(\frac{a+X_2^2}{1+X_2^2}\right)\left(\frac{1}{1+Y_2}\right){\left(1+\frac{\delta \left(4+6\delta +{\delta }^2\right){\sum }_i{X}_i}{4\left({\delta }^3+5{\delta }^2+6\delta +2\right)}\right)}^{-1}-b{X}_2,$$ (11)

$$\frac{d{Y}_2}{dt}=c-d\frac{Y_2}{1+e{X}_2^2},$$ (12)

$$\frac{d{X}_3}{dt}=\left(\frac{a+X_3^2}{1+X_3^2}\right)\left(\frac{1}{1+Y_3}\right){\left(1+\frac{{\delta }^2\left(2+\delta \right){\sum }_i{X}_i}{4\left({\delta }^3+5{\delta }^2+6\delta +2\right)}\right)}^{-1}-b{X}_3,$$ (13)

$$\frac{d{Y}_3}{dt}=c-d\frac{Y_3}{1+e{X}_3^2},$$ (14)

$$\frac{d{X}_4}{dt}=\left(\frac{a+X_4^2}{1+X_4^2}\right)\left(\frac{1}{1+Y_4}\right){\left(1+\frac{{\delta }^3{\sum }_i{X}_i}{4\left({\delta }^3+5{\delta }^2+6\delta +2\right)}\right)}^{-1}-b{X}_4,$$ (15)

$$\frac{d{Y}_4}{dt}=c-d\frac{Y_4}{1+e{X}_4^2}.$$ (16)

As expected, at $\delta =0$ the line behaves identically to the tree case, as it corresponds to a lack of coupling between cells in the network. The line network also exhibits the same solution pattern as the line in the limit as $\delta \to \infty$.

For intermediate values of δ, there are many similarities between the behaviors of the two structures, including quasiperiodicity and topology-dependent amplitudes (Fig. 5). However, there are also a key difference. Between the two extremes of δ-values, there exists a regime where the concentration of Z creates a spatial gradient based on distance from the oocyte. The closer a cell is to the oocyte, the more inhibitor it receives and the lower its amplitude as it cycles in this intermediate regime (column 2, Fig. 5).

Figure 5.

Phase planes, trajectories, and three-dimensional plots of $X_i$ for the line network case. The black line in the first row of plots shows the unperturbed limit cycle. In all plots, blue corresponds to cell 2, red cell 3, and yellow cell 4 (see Fig. 3). As δ increases, different types of dynamics can be observed. As in the tree case, the large coupling limit solution is a state where two cells are in sync and the third has a larger amplitude. To see this figure in color, go online.

Additionally, as δ increases, the cell closest to the oocyte begins to oscillate with very small amplitudes, analogous to the pattern observed in the tree network. An example of this behavior occurs when $\delta =10$ (column 3, Fig. 5). When that cell undergoes this small amplitude transition, it has a frequency of oscillation twice that of the other two cells, which oscillate in an antiphase manner. When the most proximal cell regains its amplitude, the system again goes through a complicated transition, with cell trajectories becoming quasiperiodic or chaotic. Finally, as δ increases further, its behavior becomes similar to that of the $\delta \to \infty$ limit. Overall, however, the main features of the linear network are mostly identical to those observed for the tree network.

Star network

The star network, because of its inherent structural symmetry, yields symmetric equations for the time evolution of X and Y. After simplifying the ODEs with respect to Z, we are left with the following dynamical system:

$$\frac{d{X}_2}{dt}=\left(\frac{a+X_2^2}{1+X_2^2}\right)\left(\frac{1}{1+Y_2}\right){\left(1+\frac{\delta {\sum }_i{X}_i}{2+4\delta }\right)}^{-1}-b{X}_2,$$ (17)

$$\frac{d{Y}_2}{dt}=c-d\frac{Y_2}{1+e{X}_2^2},$$ (18)

$$\frac{d{X}_3}{dt}=\left(\frac{a+X_3^2}{1+X_3^2}\right)\left(\frac{1}{1+Y_3}\right){\left(1+\frac{\delta {\sum }_i{X}_i}{2+4\delta }\right)}^{-1}-b{X}_3,$$ (19)

$$\frac{d{Y}_3}{dt}=c-d\frac{Y_3}{1+e{X}_3^2},$$ (20)

$$\frac{d{X}_4}{dt}=\left(\frac{a+X_4^2}{1+X_4^2}\right)\left(\frac{1}{1+Y_4}\right){\left(1+\frac{\delta {\sum }_i{X}_i}{2+4\delta }\right)}^{-1}-b{X}_4,$$ (21)

$$\frac{d{Y}_4}{dt}=c-d\frac{Y_4}{1+e{X}_4^2}.$$ (22)

This network structure exhibits the same patterns of solutions in the low and high δ limit as the other two cases previously analyzed. In addition, it appears that the symmetry of the topology and the equations extends to the solution structure. We found that two qualitatively different limit cycles coexist and attract different sets of initial conditions across an intermediate range of δ (Fig. 6). We also found that the sizes of the basins of attraction change as δ vary, based on the relative frequency at which each solution type emerged from time integration over initial conditions drawn from the distribution $U\left(\mathrm{0,6}\right)+0.001$ for all X and Y nurse cell components.

Figure 6.

Phase plane, trajectories, and three-dimensional plot of $X_i$ for the star network case. The black line in the first row of plots shows the unperturbed limit cycle. In all plots, blue corresponds to cell 2, red cell 3, and yellow cell 4 (see Fig. 3). As δ increases, this network first appears to yield a solution with isometric firing of each cell, followed by the same large coupling solution as the other two cases, with two cells in sync and one cell firing at larger amplitude. To see this figure in color, go online.

In the solution that attracts a larger set of initial conditions when δ is large, two out of the three nurse cells are synchronized and have a lower amplitude compared with the third cell, which additionally is phase shifted by a half period. Because any pair of cells can be the ones to synchronize, this solution corresponds to three attractors, identical in shape and related by symmetry (top, Fig. 7). This solution appears the same as the one observed in the large δ limit in all network types.

Figure 7.

Top: For all three topologies, a set of globally coupled solutions with $S_2$ symmetry exists where one cell cycles with large amplitude in $X_i$ whereas the other two cells cycle synchronously. The three-dimensional phase plane highlights the symmetry of these solutions by superimposing them on one another, with additional 2D projections. Middle: The onset of each solution with $S_2$ symmetry is dependent on the degree of coupling and the network shape. Each half line colored in blue, red, or yellow represents the range of stability of the solution where the corresponding cell has large amplitude. The green and purple segments in the star case correspond to when the two solutions with ${\mathbb{Z}}_3$ symmetry are stable. For example, in the line network, the solution where the blue cell has large amplitude is stable for $\delta \underset{\approx }{>}79$, whereas the solution where the red cell has large amplitude is stable only for $\delta \underset{\approx }{>}204.$ Bottom: The two extra solutions peculiar to the star network correspond to the isometric firing patterns that exist for an intermediate range of δ. Again, these symmetric solutions have been plotted in $\left(X_2,X_3,X_4\right)$ space, along with their projections in each direction. To see this figure in color, go online.

In the other solution, all three cells have the same amplitude and oscillate with a constant phase shift of $2\pi /3$. Although the network itself is symmetric, in defining each node with a unique label, we have introduced two possible ways that these cells may order their oscillations. These two solutions cannot be superimposed by shifting one or the other in time, resulting in two chiral attractors when observed in $\left(X_2,X_3,X_4\right)$ space (bottom, Fig. 7). We also found a regime where all three cells fire synchronously, but this regime is unstable for all values of δ.

Symmetries of globally coupled patterns

Each network model appears to have identical long-term behavior for large δ. As δ is related to the strength of coupling between cells, this large δ behavior corresponds to global coupling within the system. In this limit all solutions permit the same pattern of two synchronized cells with low amplitude and a single-cell firing at a higher amplitude with a half-period phase shift. In addition, the identity of the large amplitude cell was dependent not on the network structure but purely on initial conditions, furthering the idea that for sufficient coupling strength network topology becomes less relevant (middle, Fig. 7). Each of the three symmetrically related limit cycles corresponds to a possible identity of the cell with large amplitude (top, Fig. 7).

Although all network shapes maintain this behavior in the limit of large coupling, the symmetries and asymmetries that exist in their respective structures affect the lower bound of δ for which each of the three solutions may be observed. That is, we found that whether there exists some initial condition that makes a particular cell fire alone with a larger amplitude depends on the value of δ. For each of the three possible solutions (where either cell 2, cell 3, or cell 4 has large amplitude), parameter continuation was performed in MATCONT from an initially large value of δ to see when the solution loses stability as δ decreases (37) (middle, Fig. 7).

For each network type and for each cell, we identified the value of δ at which the globally coupled solution loses stability. For the line and tree networks, these values were found to be distinct, whereas all three solutions destabilize at the same δ in the star network. For each critical point, the system reached a limit point cycle as δ was decreased, and the stability of the solutions was lost via saddle-node bifurcation.

Overall, it is clear that for asymmetric topologies, there exists a threshold by which coupling strength dominates network shape and solutions with symmetries can exist. However, the same critical value of δ exists for all three solutions in the star network because of its inherent symmetry. In addition, another set of symmetric solutions exists for the star network outside the globally coupled solution. As δ is lowered and the large coupling solutions become unstable, we observe either one of the two limit cycles that correspond to when all three cells have the same amplitude and phase shift with respect to each other (bottom, Fig. 7). The basins of attraction for these limit cycles were found to shrink and eventually disappear as δ increases. However, for an intermediate range of δ, this network permits at least five attracting solutions.

Discussion

Our analysis of collective cell cycles in small clusters of interconnected cells demonstrates that a single parameter, which depends on the relative timescales of the intercellular transport and degradation of the cell cycle inhibitor, gives rise to many distinct behaviors based on network shape. As this parameter is varied, asymmetric systems—namely the line and tree structures—permit quasiperiodic or otherwise chaotic solutions, as well as stable limit cycles. Although no apparent symmetry is present in these two network types, the solution pattern observed for intermediate values of δ, (e.g., $\delta =10$ in Figs. 4, 5, and 6), where one cell oscillates twice as fast as the other two antiphase cells, is characteristic of a symmetric coupled network. Specifically, in a coupled system of three cells exhibiting ${\mathbb{Z}}_2$ symmetry (arranged in a line), it has been previously shown that one possible solution to the corresponding dynamical system is one where the middle cell oscillates with twice the frequency of the other cells which are half a period phase shifted (21). However, in this case all cells are assumed to be identical, and it is simply the placement of the middle cell between two others that allows this symmetry. In the cases we analyze here, the system itself is asymmetric, but still permits symmetric solutions based on the value of δ. That these solutions are observed for larger values of δ implies that some form of symmetry is introduced into the line and tree systems as coupling strength increases.

In addition to the symmetries found in the dynamics of asymmetric networks, we also observe in the large δ limit that all three networks have solutions with $S_2$ symmetry, where two arbitrary cells are synchronized and half a period out of phase with the third cell. We hypothesize that this homogeneity of solution pattern across networks stems from a global coupling effect: each supporting cell receives an identical signal because of the rapid accumulation and redistribution of the inhibitor, which renders the specific network topology irrelevant. This type of solution has also been shown to exist for the same three-cell linear network described above (21).

By contrast, the structure of the star network yields inherent $S_3$ symmetry. This symmetry is preserved in the unstable synchronized solution and broken in the two types of solutions which are both stable for intermediate δ. These findings are consistent with previous work on a “ring” of three oscillators, where the synchronous solution was termed “in-phase” and the two solutions with broken symmetry “rotating wave” and “2-in-phase.” The latter two correspond to solutions with ${\mathbb{Z}}_3$ and $S_2$ symmetry, respectively (38). Our results further confirm that the effects of the exact forms of equations on system dynamics are secondary to the constraints imposed by the underlying symmetries of the coupled cell system.

Although our model is clearly a simplification of the comprehensive molecular and cellular dynamics in real germline cysts, it provides insights into the types of collective behaviors within real biological systems. For example, in the line network, we found that oscillation amplitude decreases as the distance to the oocyte decreases, due to the increased effect of the diffusing inhibitor. This observation (column 2, Fig. 5) is consistent with the hierarchy found in the Drosophila egg chamber (17), where cells the same number of connections away from the oocyte appear to have similar amplitudes and frequencies and thus form oscillation groups.

This finding is even more clear in the tree network, where the two cells directly connected to the oocyte have similar amplitudes that are smaller than that of the remaining cell (column 2, Fig. 4). This captures the idea of growth coordination groups defined based on the connectivity patterns of group members (17). Finally, as indicated in the star topology, symmetric networks give rise to symmetric branches. For biological systems comprised of symmetric branches (e.g., D. cuprea, Fig. 1 B), one would expect each linear branch extended from the oocyte to exhibit the same pattern of limit cycle oscillations.

As our scan of δ shows an array of different behaviors across a wide range of values, it is worth looking deeper into what value of δ corresponds to the real properties of the diffusive inhibitor we analyze. Previous work has established the rate of diffusion of human p27, as well as the rate of p27 degradation (39,40). Although recent results showed the diffusion rate in humans was analogous to its rate in Drosophila, further work is needed to measure these rates in Drosophila, as well as other model systems, using tools such as fluorescence recovery after photobleaching or photoconversion methods (17,41).

Although the work of this article highlights general features of network topology in driving various behaviors, real biological systems may be more sensitive to the presence of cell cycle components. All in all, we expect future studies of collective dynamics in cell clusters with stable cytoplasmic bridges will test our predictions about the interplay of network structure and intercellular transport in this important and interesting class of small multicellular systems.

Author contributions

S.Y.S. designed the research. B.S. and R.D. performed analytical and numerical studies.

Editor: Alex Mogilner.