Self-sustained collective oscillation generated in an array of nonoscillatory cells

Oscillations are ubiquitous phenomena in biological systems. Conventional models of biological periodic oscillations usually invoke interconnecting transcriptional feedback loops. Some specific proteins function as transcription factors, which in turn negatively regulate the expression of the genes that encode these “clock proteins.” These loops may lead to rhythmic changes in gene expression in a cell. In the case of multicellular tissue, collective oscillation is often due to the synchronization of these cells, which manifest themselves as autonomous oscillators. In contrast, we propose here a different scenario for the occurrence of collective oscillation in a group of nonoscillatory cells. Neither periodic external stimulation nor pacemaker cells with intrinsically oscillator are included in the present system. By adopting a spatially inhomogeneous active factor, we observe and analyze a coupling-induced oscillation, inherent to the phenomenon of wave propagation due to intracellular communication.

Figure 1

Profile of ${\Gamma }_i$ obtained from Eq. (4), when ${\Gamma }_0=15,\xi =5$.

Figure 2

Nullcline diagrams in (a) bistability and (b) monostability, respectively.

Figure 3

Collective oscillation observed in a chain of cells when $\tilde{D}=0.7$ in Eqs. (8, 9). (a) Spatiotemporal plot of the collective oscillation of $u_i$. The black and white indicate $u_i=1$ and $u_i=0$, respectively. [(b) and (c)] Wave forms of $u$ (solid) and $v$ (dashed) in the 3rd and 4th cells.

Figure 4

(a) Spatiotemporal diagram of $u_i$ over a single period. [(b)–(f)] Snapshots of $u$ and $v$ at several time points in one period, where the horizontal axis is cell number from 1 to 20. This illustrates the change in wave propagation at different stages. Solid curves and dashed curves indicate $u$ and $v$, respectively. An animation, through which the behavior can be understood more intuitively, is available at [54].

Figure 5

Spatiotemporal diagram of $u_i$ in a stationary state. Wave propagation stops and no oscillation occurs in the case of weak coupling $\tilde{D}=0.47$.

Figure 6

Antiphase mode in period doubling produces collective oscillation with a periodic position shift when $\tilde{D}=0.8$. (a) Spatiotemporal diagram of $u_i$. The grayscale black and white indicate $u_i=1$ and $u_i=0$, respectively. (b) and (c) are wave form diagrams of the 3rd and 18th cells. Activator $u$ and inhibitor $v$ are shown in solid and dashed curves, respectively.

Figure 7

In-phase mode in period doubling produces collective oscillation with a change in the periodic population when $\tilde{D}=0.9$. (a) Spatiotemporal diagram of $u_i$. The grayscale black and white indicate $u_i=1$ and $u_i=0$, respectively. (b) and (c) are wave form diagrams of the 3rd and 18th cells. Activator $u$ and inhibitor $v$ are shown in solid and dashed curves, respectively.

Figure 8

Spatiotemporal diagram of $u_i$. Black and white indicate $u_i=1$ and $u_i=0$, respectively. (a) $\tilde{D}=2.53$, oscillation starts to collapse; (b) $\tilde{D}=2.6$, oscillation ceases after one cycle.

Figure 9

Phase diagram of (a) cell number for the left boundary of collective oscillation. (b) The oscillation period, (c) an enlarged view with in-phase and antiphase period-2 solutions, with respect to the change in the strength of coupling.

Figure 10

Spatiotemporal diagram of $u_i$. Black and white indicate $u_i=1$ and $u_i=0$, respectively ($N=200$). (a) $\tilde{D}=D/\Delta x^2=0.7$. (b) $\tilde{D}=D/\Delta x^2=70$.

Figure 11

Phase portrait diagrams with snapshots of the dynamical nullcline. Rows indicate the time evolution from the top down and columns indicate the number of cells (3 at left, 4 at middle, and 5 at right). Dashed curves are the limit cycle solution. Cubic function curves are the nullcline of ${\dot{u}}_i=0$. Straight lines are the nullcline of ${\dot{v}}_i=0$. Circles are the position of $\left(u_i,v_i\right)$ at specific times. An animation of the dynamical nullclines can be found at [54].

Figure 12

Schematic diagram of two critical tangency situations, corresponding to the conditions for which (a) a wave back passes and (b) a wave front is generated. Black circles are the position of states when two nullclines tangent to each other.

Figure 13

Variation of two tangent points $u_{T_{1,2}}$ with respect to $\Gamma$.

Figure 14

The diagram in the parameter plan $\left({\Gamma }_c,\tilde{D}\right)$ representing the conditions of collective oscillation. A wave back passes the cell and a wave front reflects, when the parameters are above the WB and WF line, respectively.