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Figure 1. Cortical waves of Rho activity in Patiria miniata oocytes. Here oocytes have overexpressed Rho-GEF Ect2, induced expression of ∆90 cyclin B, and are treated with roscovitine. Both the experiment31 and our model show similar overall dynamics and the presence of wave domains. (A) Snapshots of wave propagation in the experiment. The images are generated from the data published in Bement et al.31 with the permission of the authors. The scale bar is 50 μm. (B) Snapshots of wave propagation in our simulation (ð‘˜0=0.2, ð‘ 2=1, ð›¼1=3, ð›¼2=0.1). The cyan contours outline wave domains as defined in the following section, while the arrows indicate the direction of wave propagation in these domains. For the full dynamics see Supplementary Video 16 in Ref.31 and our Supplementary Video S1.
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Figure 2. The model and an illustration of the wave domain concept. (A) Diagram of the signaling motif. ð´ and ð¼ represent active and inactive forms of the signaling molecule, while ð¹ represents a local inhibitor (see âMethodsâ for equations and Table 1 for parameter values). (B) The resulting distribution of component ð´ in a snapshot of the simulation with ð0=0.15, ð 2=0.8 using homogeneous initial conditions. (C) An image of two merged channels: green channel represents activator ð´ and red channel represents inhibitor ð¹. The concentrations were scaled as [ð¶âððð(ð¶)]/[max(ð¶)âððð(ð¶)]. (D) The colormap of the direction angles, ð, of the wave vectors at the same time point as B and C. Arrows represent the direction of wave vectors. (E) The colormap of the magnitude of a local change (gradient) in the directions of the wave vectors. Regions of high gradient are located at the edges of wave domains. (F) The colormap of our metric of the coherence distance. (G) Segmentation of the coherence distance with the modified watershed algorithm and after merging domains with similar values of the mean ð. (H) The dynamics of wave domains. The number of regions detected by the automated segmentation pipeline is decreasing (left panel), while the mean area of regions is increasing over time (right panel).
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Figure 3. Parameter scan and textural analysis of the emerging patterns. (A) The result of model simulations for a range of parameters ð0 and ð 2 (see also Supplementary Video S3). (B) The colormap of the excitation measure (see âMethodsâ) for the parameter space in (A). (C, D) The textural measures of pattern entropy and correlation for the parameter space in (A). Each measure was averaged over 100 simulation steps after the formation of waves from the initial homogeneous state.
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Figure 4. Two stages of pattern development and characteristics of the regimes in the low-activity stage. (A,B) The amplitude of the activator pattern, maxð‘¥,ð‘¦ð´âˆ’minð‘¥,ð‘¦ð´, during the low- and high-activity stages for the quasi-static low-activity regime (ð‘˜0=0.2, ð‘ 2=0.8) and the oscillatory low-activity regime (ð‘˜0=0.2, ð‘ 2=1.1), respectively. (C,D) The same plots as in (A), (B), but in the log scale. (E,F) A series of snapshots of the system behavior from the same parameters as in (A,C) and (B,D), respectively (see also Supplementary Videos S4–S6). The concentrations were scaled as [ð¶âˆ’minð¶]/[maxð¶âˆ’minð¶] both for the inhibitor (red channel) and the activator (green channel) before the channels were merged together. (G,H) (Left panels) Snapshots of the system in the quasi-static (ð‘˜0=0.2, ð‘ 2=0.8) and oscillatory (ð‘˜0=0.2, ð‘ 2=1.1) regimes of the low-activity stage; top and bottom panels respectively. (Central panels) Kymographs of the system dynamics along the vertical lines shown in the left panels. For comparison of patterns with different amplitude, the concentration values were scaled to zero mean and standard deviation equal to one (mean value was subtracted from each pixel and then divided by standard deviation). (Right panels) Temporal autocorrelation plots, showing the correlation coefficient as a function of the time lag (see “Methodsâ€). (I) Classification of the parameter space with white and red colors indicating the quasi-static and oscillatory regimes, respectively. (J) Colormap showing the duration of the low-activity stage (in s).
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Figure 5 The transition of the system between the low- and high-activity stages. (A) Snapshots of the patterns at low- and high- activity stages for different values of the inhibition strength: (1) weak (ð0=0.18 , ð 2=0.7 for quasi-static and ð0=0.2, ð 2=0.75 for oscillatory regimes), (2) balanced (ð0=0.2, ð 2=0.9 for quasi-static and ð0=0.26, ð 2=1.05 for oscillatory regimes), and (3) strong (ð0=0.2, ð 2=1.1 for quasi-static and ð0=0.2, ð 2=1.1 for oscillatory regimes). (B) Snapshots of the system at the beginning of the high-activity stage. Arrows indicate the direction of wave propagation. Dashed outlines indicate vector alignment on a larger scale. Images in (A), (B) show two merged channels: green for the activator and red for the inhibitor. (C) The activity amplitude and temporal autocorrelation for the oscillatory low-activity stage with weak negative feedback (left), balanced activation and inhibition (center), and strong negative feedback (right).
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Figure 6. Classification and long-term dynamics of wave domains. (A) Different types of systemâs behaviors in the space of parameters representing the basal activation rate and the strength of negative feedback. (B) Textural entropy as a function of time during the growth of wave domains (ð0=0.15, ð 2=0.8). (C) Snapshots from the simulation in B with the simultaneous visualization of the inhibitor and activator in the system (merged channels). (D) Colormaps of the direction of wave propagation in (C) showing the increasing size and decreasing number of the wave domains over time. The values of wave vector angles were averaged with a running window of 83 s.
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Figure 7. Wave domains analysis for experimental starfish data. (A) The pipeline of wave domain identification in starfish experimental images, including active Rho pattern, colormap of wave vector directions, colormap of the coherence distance, and wave domain segmentation. The scale bar is 50 μm. The image in the left panel is generated from the data published in Bement et al.31 with the permission of the authors. (B) The mean size of wave domains as a function of time for starfish oocyte. The grey area indicates the period of time after the sharp switch in the cell behavior.
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