Random cell movement promotes synchronization of the segmentation clock

Abstract

In vertebrate somitogenesis, the expression of segmentation clock genes oscillates and the oscillation is synchronized over nearby cells. Both experimental and theoretical studies have shown that the synchronization among cells is realized by intercellular interaction via Delta-Notch signaling. However, the following questions emerge: (i) During somitogenesis, dynamic rearrangement of relative cell positions is observed in the posterior presomitic mesoderm. Can a synchronized state be stably sustained under random cell movement? (ii) Experimental studies have reported that the synchronization of cells can be recovered in about 10 or fewer oscillation cycles after the complete loss of synchrony. However, such a quick recovery of synchronization is not possible according to previous theoretical models. In this paper, we first show by numerical modeling that synchronized oscillation can be sustained under random cell movement. We also find that for initial perturbation, the synchronization of cells is recovered much faster and it is for a wider range of reaction parameters than the case without cell movement. When the posterior presomitic mesoderm is rectangular, faster synchronization is achieved if cells exchange their locations more with neighbors located along the longer side of the domain. Finally, we discuss that the enhancement of synchronization by random cell movement occurs in several different models for the oscillation of segmentation clock genes.

Fig. 1.

Model scheme for the segmentation clock in vertebrate somitogenesis. (A) The presomitic mesoderm (PSM) and the two-dimensional lattice representing the posterior part of the PSM. The purple area indicates the expression of the her gene. (B) Exchange of location between two cells. (C) Each cell exchanges its location with one of its two neighbors in the long-side direction of the lattice with probability p_l/2, and it exchanges its location with one of its two neighbors in the short-side direction of the lattice with probability (1 - p_l)/2. (D) Negative feedback regulation of the her gene in a cell and intercellular interaction between neighboring cells via Delta–Notch signaling. Production, activation, and transport are represented by red arrows, whereas suppression is shown by blue lines ending with perpendicular bars. Decay of mRNA and proteins is denoted by arrows leading to symbol Ø. (E) Schematic representation of how an initial condition was prepared. The circle drawn with the broken line represents the limit cycle with all the N cells perfectly synchronized in the phase space of m, y, z, and w (Appendix). Initial states of cells are distributed in the shaded region that wraps a part of the limit cycle (see Model and Appendix for details).

Fig. 2.

Random cell movement enhances the restoration of synchronization. (A) Correspondence between 3D plots of her mRNA concentration and density plots of that. In the density plots orange color indicates high concentration of her mRNA, and blue indicates low concentration. (B), (C) The time courses of her mRNA expression with cell movement (Right Columns; indicated by “move”) and without cell movement (Left Columns; indicated by “no move”) between (B) 124 min and 180 min, and (C) 268 min and 324 min. In right columns of (B) and (C), each cell experienced an exchange of its location with one of its neighbors within 10 min on average. (D) The time courses of her mRNA expression in 15 of N cells when all N cells are fixed in the lattice [Upper; corresponding to left columns in (B) and (C)] and when the cells moved randomly in the lattice [Lower; corresponding to right columns in (B) and (C)]. The 15 cells were chosen randomly among N cells. (E) The time courses of IS corresponding to (D). (F)–(H) The average time course of IS when the magnitude of the initial phase differences between cells was set to (F) α = 0.6, (G) α = 0.8, and (H) α = 1.0. Each cell experienced an exchange of its location with one of its neighbors within 5 min (Red Filled Circles), 10 min (Green Open Circles), or 20 min (Blue Filled Squares) on average. Pink open squares represent the case in which all N cells were fixed in the lattice. We used 20 different initial conditions. We ran 100 simulations for each initial condition in which we considered cell movement. We averaged all of the trials (2,000 runs for systems with cell movement and 20 runs for systems with cells fixed in the lattice). Error bars indicate standard deviations. The reaction parameter values used in Eq. 2 are listed in the Appendix as the standard parameter set.

Fig. 3.

Cell movement extends the parameter range that allows cells to maintain synchronization. (A) The time courses of the average IS when the threshold constant for the suppression of her mRNA by Her protein K₁ = 0.6. Other parameters were set to the standard values listed in Appendix. Each cell experienced an exchange of its location with one of its neighbors within 10 min on average (Green Open Circles). Pink open squares represent the case in which all N cells were fixed in the lattice. The magnitude of the initial phase differences between cells was set to α = 0.2. Error bars indicate standard deviations. (B), (C) The average IS after 10 cycles for different values of the threshold constant for the suppression of her mRNA by Her protein (K₁) and that for the suppression of Delta protein by Her protein (K₇). Other parameters were set to the standard values listed in Appendix. (B) Cells were fixed in the lattice, or (C) each cell experienced an exchange of its location with one of its neighbors within 10 min on average. The magnitude of the initial phase differences between cells was set to α = 0.6. In (A–C) we used 10 different initial conditions for each parameter set. We ran 10 simulations for each initial condition when we considered cell movement. Then we averaged all trials (100 runs for systems with cell movement and 10 runs for systems with cells fixed in the lattice).

Fig. 4.

The aspect ratio of a lattice determines the optimal magnitude of anisotropy in the direction of cell movement for quick recovery of synchronization. (A)–(D) Average IS over 2,000 simulations of 5, 10, and 20 cycles when each cell exchanged its location with one of its neighbors with each p_l in a (A) 25 × 10 lattice, (B) a 16 × 16 lattice, (C) a 32 × 8 lattice, and (D) a 64 × 4 lattice. In (A)–(D), each cell experienced an exchange of its location with one of its neighbors within 5 min on average. The magnitude of the initial phase differences between cells was set to α = 1.0. We used 20 different initial conditions and ran 100 simulations for each initial condition. Error bars indicate standard deviations. Reaction parameter values in Eq. 2 were the standard parameter set listed in Appendix.