#2005-10-12355C Horikawa Supplementary Notes S3.
Supplementary Notes S3: Description of a one-dimensional simulator
To understand the collective oscillation pattern in numerous cellular oscillators, we used one mathematical model originally described by J. Lewis. Detailed mathematical analysis with the single-cell and two-cell system has been reported in ref S16. For simulation performed in this report, we simply expand the Lewis’s system into one dimensional array of multiple cells.
Delay model (summary of the Lewis’s report)
In the delay model, the quantity of protein and mRNA for her1, her7 and deltaC in each cell are supposed to obeys the following equations.
p(t) and m(t) represent the number of the mature protein and mRNA in each cell at time t. a, b and c is a protein synthesis rate per mRNA molecule, protein degradation rate and mRNA degradation rate, respectively. ‘delayT’ is a fixed time necessary for the maturation (synthesis, modification and translocation) of each molecule. This delayT plays a key role both in generating sustained oscillation and in determining an oscillation periodS16.
Transcription of oscillating genes (her1, her7 and deltaC) is considered as a function of , and ( denotes the value of in neighboring cells). HER1/HER7 and DELTA are supposed to control the gene expression as a hetero-dimmer and monomer, respectively. The transcription rates f are represented as follows,
‘r’ is a weight parameter for each regulation, assumed to add up to 1. denotes the reduced value of each protein concentration divided by critical value that is the critical number of molecules for the regulation of target gene transcription. ‘k’ is a maximal transcription rate in the absence of regulation (for details, see ref S16).
Simulation 1: The phase-shift induced by the transplantation
To test the phase-shift effect induced by the explants that constitutively expresses the Delta protein, we calculate the coupled oscillation of linearly aligned ten cells. Delta protein expressed in a cell is supposed to influence both neighboring cells (Box 1).
Equations
Equations and parameter values used in this simulation are,
Here, [ ]t-T denotes the value of the expression in the square bracket evaluated at the time point t – T, that is delay time.
Parameters are identical in all cells.
Here, we assumed that transcription of her1, 7 is a consequence of both Her dependent repression and combinatorial regulation by Her and Delta, while that of Delta is solely regulated by Her in each cell (for simplicity, basal and Delta-dependent regulation is ignored). Thus, the transcription of oscillating genes in cell n is represented as follows,
Here, , , (n) represents the position of the cell. and are the weight parameter that determine the balance of internal and external contribution of oscillating molecules. Thus, we define the strength of coupling as rhd/rh. ()
Other parameters are,
Initial condition
At t=0, concentrations of all components are set to zero.
Boundary condition
The posterior-most cell (n=10) is assumed to constantly receive the high amount of DeltaC stimulation (DeltaC=500) from the border (transplanted) cell.
The anterior-most cell (n=1) is supposed to receive the DeltaC stimulation the amount of which is identical to that released by itself.
Results in simulation 1
In this 1-D simulation, 10 linearly connected cells along the A-P axis are represented by each dot. In the normal condition, the levels of her1 mRNA oscillate synchronously, as seen in vivo (light gray dots in Fig. 1hFig. S6a).
We next examined the response of normally oscillating cells to external stimuli by placing a continuously signaling cell at the posterior end of the virtual PSM. This hypothetical signaling cell (Red arrow in Fig. S6ah) expresses high levels of Delta which constitutively activate Notch receptor in the immediately adjacent PSM cell. During several rounds of oscillation, the synchrony is gradually affected in the region near the actively signaling cell (gray dots in Fig. 1hFig. S6a, see also Supplementary Movie S1). Moreover, the oscillation dynamics of each cell, traced as a function of time, reveals that the oscillation period becomes shorter in cells that are in proximity to the active cell, leading to accelerated oscillation in comparison with the non-affected control region (Fig. 1iFig. S6b).
These results are highly consistent with those obtained by the above in vivo experiments, supporting the idea that the segmentation clock behaves as a coupled oscillator, in which cellular oscillators are inter-connected through the oscillator-linked Notch signaling.
These results both in in vivo and in silico indicates that the segmentation clock actually behaves coupled oscillator based on the proposed molecular circuits.
Fig. S6. Simulated effect of actively signaling cells on the synchronized oscillation.
a, 1-D simulation of oscillating PSM cells (dots). Snap shots of the calculated results in the 10th-round of oscillation (blue inset in b) are shown. Actively signaling cell is represented as red arrow. See also Supplementary Movie S1. b, Time course of her1 oscillation in affected and non-affected cells (cell B and cell A in a).
We have also examined the dependence of the phase-shift effect on the value of coupling strength, rhd/rh. As shown in Fig. S7, strong intercellular coupling (rhd/rh>10) is required for effective phase-shift. Thus, we adopt (rh, rhd)=(0.05, 0.95) in above described simulation.
Fig. S7. Phase acceleration depends on coupling strength.
The phase differences between cell A and cell B (see Fig. S6) at t=500 (y-axis) are plotted against the strength of intercellular coupling (x-axis).
Simulation 2: Synchronized oscillation in the presence of noise
We simulate the effect of noise on the collective oscillation with a set of 25-cell array (Box 1). Maturation of the macro-molecules (protein and mRNA) is energy dependent process. Thus, it should be expected that the delay time is slightly different among cells reflecting stochastic gene expression and their different micro-environment. As the delay time mainly determines the frequency of the oscillationS16, this type of noise causes the randomization of the oscillating phase in the absence of coupling.
Another cause of noise is cell proliferation that occurs in numerous cell in the PSM (Fig. 4a-b, Supplementary Movie S2 and Supplementary Table S1). During the mitotic process, it is expected that the maturation of the molecules is transiently slowed down.
The effects of noise described above, are calculated in the simulation 2 (Box1) as follows.
Effect 1; different frequency in each oscillator
1) Delay time for all components in each cell is distributed with the standard deviation value of 0.15 and is fixed throughout the simulation.
2) Mean of the delay time is,
Effect 2; phase-shift associated with mitosis
1) Dividing cells are randomly selected every minute with 0.2%. This produces the condition that 7% of cells in a total population experience mitosis during one cycle of oscillation, which is consistent with our experimental observation. (Fig. 4a, Supplementary Table S1)
2) Each division is supposed to last for 15 min (Fig. 4b).
3) During the division, the delay time is doubled and the synthesis rate of mRNAs and proteins is set to the halves.
Parameters
The pattern of collective oscillation in the presence or absence of intercellular coupling was calculated for the value of rhd/rh is larger than 10 () and is smaller than 0.1 (), respectively (see also Fig. S7). Values of other parameters are identical to that of simulation 1
Simulation 3: synchronization by the intercellular coupling
To test the phase synchronization of the segmentation clock, we calculated the effect of phase-delayed explants in the cell array (N=15) using the deterministic model.
Virtual transplantation is performed as follows. After the uniform and stable oscillation is achieved in the array of fifteen cells, two cells at the center of them are replaced with those oscillating out of phase (delayed by 1/4-cycle) at iteration time 140 min. Initial conditions and parameters are identical to those in simulation 1. The coupling between the host and transplanted cells are assumed to be identical to that of normal host cells.
We found that the phase of the explants is easily adjusted to the host phase without affecting the host oscillation (Fig. S8 and Supplementary Movie S3). This is one of the most characteristic features of coupled oscillator systems, designated as ‘phase synchronization’, observed when two or more non-linear oscillators are coupled.
Fig. S8. Phase synchronization in the segmentation oscillator.
a, Interaction between two oscillating groups calculated by a 1-D simulator. Two cells with delayed phase (1/4-cycle) are transplanted at t=140min (blacket). The phase of the explants is gradually synchronized to that of the host in several rounds of oscillation. b, The levels of her1 mRNA in host (cellA, green) and explants (cellB, magenta) are traced as a function of time. See also Supplementary Movie S3.
We also examined the dependence of this effect on the coupling strength, and found that the synchronization effectively occurs when the value of rhd is larger than 0.6 (Fig. S9).
Fig. S9. Coupling-dependence in the phase synchronization.
In simulation 3, time-development of phase difference between cell A and cell B (Fig. S8a) at t=500 (y-axis) is calculated for each coupling strength. Phase difference at the transplantation (25% at t=140) is efficiently reduced in the presence of intercellular coupling, rhd>0.6
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