The GFP-fused pleckstrin-homology domain of Akt/PKB (PHAkt/PKB) was expressed in wild-type AX-2 cells. Cells were cultured at 21°C in HL-5 medium and selected with 20 μg/mL G418 (Watts and Ashworth, 1970). Before observation, the cells were placed in glass bottom dishes (IWAKI, Japan) 27 mm in diameter and starved in development buffer (DB: 5 mM Na phosphate buffer, 2 mM MgSO4, and 0.2 mM CaCl2, pH 6.3) for 4.5 h, leading to chemotactic competency with respect to cAMP (Arai et al., 2010). The glass bottom dishes were used in the presence or the absence of polyethylenimine (PEI) at a concentration of 2.4 μg/mL in DB. The number of cells was maintained at less than 5±10⁵ cells to sufficiently separate neighboring cells. After starvation the extracellular fluid was exchanged and the cells were incubated for an additional 20 min incubated before observation with 1 mL DB supplemented with 20 μM latrunculin A (L5163-100UG, Sigma), which inhibits actin polymerization, and 4mM caffeine, that inhibits the secretion of cAMP (Brenner and Thoms, 1984). A total of 79 cells were observed on glass and 98 cells on PEI-coated glass. The latter increases the adhesion strength of cells and leads therefore to a larger fraction of the plasma membrane adhering to the substrate (ventral membrane) (Hörning and Shibata, 2019).

Fluorescence images were obtained using an inverted microscope (IX81-ZDC2; Olympus) equipped with a motorized piezo stage and a spinning disc confocal unit (CSU-X1-A1; Yokogawa) through a 60x oil immersion objective lens (N.A. 1.35; UPLSAPO 60XO). PH_Akt/PKB-EGFP was excited by 488-nm laser diode (50 mW). The images were passed through an emission filter (YOKO 520/35, Yokogawa) and captured simultaneously by a water-cooled EMCCD camera (Evolve-, Photometrics). Time-lapse movies were acquired at 10-s intervals at a spatial resolution of dx = dy = 0.2666 μm and dz = 0.5 μm using z-streaming (MetaMorph 7.7.5), which enables recording the entire focal volume in about 3s.

The recorded spatiotemporal fluorescence signaling of cells was reconstructed considering the entire four-dimensional data set (x, y, z and t). The cell surface was represented by a triangular mesh that was generated using a Delaunay triangulation routine in MATLAB (Mathworks) (Persson and Strang, 2004), and transformed to a volume filling three-dimensional tetrahedral mesh by connecting the surface triangles to the center coordinate of the cell. The sections were evaluated by the time-averaged, radial intensity profile of all pixels within one grid element. The maximum intensity of the time-averaged standard deviation determined the membrane position. Thereafter, the position of nodes is determined, so that the average node area was approximately 0.3 μm². Finally, the grid nodes were smoothed among neighboring nodes using a triangular low-gaussian kernel. The PIP3 domain dynamics were quantified by spherical harmonic expansion to the 3rd order (octupole moment), which smoothened the intensity distribution for peak detection of the domains. The domain velocity, v is calculated from the temporal peak displacement on the membrane by approximating the peak-to-peak distance as the shortest distance along a straight line on the membrane. A more detailed methodological description can be found in Hörning and Shibata (2019). The generated data tables used for this study can be found in the Supplementary Material.

The two-dimensional homolographic equal-area Mollweide projection is used to visualize membrane signaling in 2D. The transformation from spherical coordinates (ϕ, θ) to cartesian coordinates (x, y) is performed through the equations

with the auxiliary angle γ given as

which can be solved iteratively by the Newton-Raphson method (Snyder, 1987; Hörning and Shibata, 2019).

The statistical self-affinity of the extracted PIP3 signaling on the membrane of Dictyostelium cells was determined by Detrended Fluctuation Analysis (DFA) of first order (Peng et al., 1994). A previously introduced scheme in Matlab was used to determine the scaling exponent, α from the time series (Habib, 2017; Habib et al., 2017). The cumulated sum S(t_i) of the discrete signal s(t_i) is computed as

where ŝ denotes the mean of s(t_i). The variance F²(τ) is determined within the time window T_max = 180 s by obtaining the best linear fit P of the cumulated sum S, as

Plotting the τ against F²(τ) on a log-log graph indicates α as the linear slope, since F(τ)~τ^α. The extension of this framework for higher orders (i.e., DFAn) is explained elsewhere (Habib, 2017; Habib et al., 2017). Examples of different numerically generated and analyzed time series are shown in Supplementary Figure 1.

Phase information was extracted from the raw signals using the WAVOS toolkit for wavelet analysis and visualization of oscillatory systems in Matlab (Harang et al., 2012). The complex Morlet wave function

is applied to the PIP3 fluorescence intensity data at each membrane node using ω = 2π in the range of t_min = 60 s to t_max = 330 s. The dominant frequency in the power spectra was determined as the dominant frequency mode. The phase was calculated from the complex wavelet coefficients of the dominant frequency mode. Finally, all extracted phases were reconstructed and mapped on the membrane mesh.

In this study, actin polymerization-inhibited Dicyostelium cells were observed and analyzed by taking advantage of the simple and symmetric cell shape. For that cells were treated with 10 mM latrunculin A and 4 mM caffeine to suppress cell motility, formation of membrane protrusions, and cell-cell interactions (Arai et al., 2010). Thus, the cell membrane can be described and mapped using a spherical coordinate system with the polar angle θ and the periodic azimuthal angle ϕ (Figure 1A). Further, the cell shape was described by the contact angle Ω and the area of the ventral membrane, i.e., the area of the membrane that adheres to the substrate, A_adh.

Due to the simple symmetric shape of the cells, it is also possible to project the three dimensional PIP3 signal activity onto a two dimensional map, as shown in Figure 1B. The upper two panels show the front and back side of a cell with irregular complex shaped PIP3 domains, and the lower panel shows the corresponding 2D map (Mollweide projection; see Materials and Methods) of the entire membrane signaling. For that a customized mapping routine was applied, as introduced in detail before by Hörning and Shibata (2019). Briefly, the temporal fluorescence variation of the signaling at each local position was used to specify the membrane position, from which a three-dimensional triangular map was constructed using a Delaunay triangulation routine (Persson and Strang, 2004) with a constant mesh size of approximately 0.3 μm². Depending on the cell size, this approach leads to membrane grids with 620–6,632 nodes, i.e., triangles. Additionally, part of the cells were cultured on polyethylenimine (PEI) coated glass bottom dishes to increase the adhesiveness of cells to the glass substrate and therefore having a wider range of different cell shapes. The ventral membrane area increased from A_{Adh, glass} = 35.6 ± 5.6 μm² to A_{Adh, PEI} = 51.9 ± 4.2 μm² (mean ± SE). Between the two different substrates no influence on the PIP3 signaling was observed.

Using this methodological approach the three basic PIP3 domain dynamics could be observed. The horizontal waves that propagate parallel to the contact perimeter (Figure 1C), vertical waves that periodically cross the contact perimeter (Figure 1D), and transient spot waves that are small domains that appear and disappear repeatedly at seemingly random positions at the membrane above the contact perimeter (Figure 1E). The corresponding movies can be found in the Supplementary Material.

Figure 2A shows kymographs of the three PIP3 domain dynamics observed on Dictyostelium membranes: horizontal waves, vertical waves, and transient spot waves. While those wave patterns have been analyzed by tracking only the domain center using spherical harmonic analysis (Hörning and Shibata, 2019), the raw PIP3 signaling reveals irregular complex shaped domains (Figures 1B–E). Taking advantage of the high spatial resolution of the triangular mesh, it is possible to analyse the PIP3 signal at each triangle individually. Figures 2B,C show two typical periodic and non-periodic time series observed at single locations on different cells. Those signals can be independently analyzed by DFA. The resulting slopes enable the quantification of signal- and noise-dominant regimes in the range of α = [0, 2], where α = 0.5 corresponds to white noise and α = 2 to a signal without noise (see also Supplementary Figure 1). PIP3 signaling on Dictyostelium membranes was observed in the range of α ≃ [0.5, 1.5] similar to the shown periodic (α~1.2) and non-periodic (α~0.6) examples in Figures 2B–D.

Mapping α on the entire membrane (Figures 2E–G) reveals the underlying spatiotemporal pattern formations (Figure 2A). The yellow (α>1) and blue (α < 1) colors distinguish signal- and noise-dominant spatial locations on the membrane, respectively. The dashed line marks the contact perimeter. The right panels show the probability distribution of α along the vertical cell-symmetry axis from θ = −π/2 (cell bottom) to θ = +π/2 (top of cell) with the contact perimeter marked as a dashed line. Generally, we observed that smaller cells and those with a small ventral membrane area show more uniform distibutions of constant α (Figure 2E). Larger cells with a larger ventral membrane showed dominant periodic activity on the ventral membrane (Figure 2F), and smaller cells showed dominant periodic (yellow) activity at the upper plasma membrane with noisy activity below the contact perimeter (Figure 2G). The latter can be observed in very sharp transitions of activity above and below the contact perimeter (Figure 3A), implying an active role of the contact perimeter in PIP3 domain activity. Contrarily, the cell shown in Figure 3B shows constant values of α below the contact perimeter, and a gradual decrease of α toward the top of the cells (θ = π/2).

Figures 3C,D show time-series snapshots of PIP3 signaling corresponding to the two representative α-distribution maps (Figures 3A,B). Figure 3C illustrates how the contact perimeter blocks the domain motion toward the ventral membrane of the cell, which leads to domain guidance along the contact perimeter. The difference of domain activity gets more clear when comparing the kymographs above and below the contact perimeter (right panel). The latter shows no domain motion. However, in larger cells (Figure 3D), we observed that the domains can pass the contact perimeter and lead to domain motion on the entire membrane.

In order to quantify the role of the contact perimeter in membrane signaling, a total of 178 cells were statistically analyzed with focus on the spatial distribution of α, as shown in Figures 3A,B (right panels). The weighted polynomial fits (black solid line) of the α distributions were parameterized by α_min and α_adh, which define α at the polar angles θ_min=−π/2 at the bottom of the cell and θ_adh at the spatial position of the contact perimeter, respectively (Figure 4A). Further, we introduce the scaling parameter Δα = α_adh − α_min, which parameterizes the difference in signal activity between the bottom of the cell (θ_min) and the contact perimeter (θ_adh). A negative Δα is associated with a highly periodic or oscillatory signal at the ventral membrane (Figures 2F, 4A, upper black line), while a positive value indicates noise-dominant signaling at the ventral membrane (Figures 2G, 4A, lower black line). Figure 4B shows the relation between α_adh and α_min, where the color scheme indicates cells of different α_min (see also Supplementary Figure 3). This color scheme provides an additional fingerprint of the membrane signaling independently from the shape of the weighted polynomial fits (Figure 4C). Cells that show Δα ≃ 0 are cells of comparable signaling at the center of ventral membrane and contact perimeter (α_min ~ α_adh). Those cells are either dominated by periodic PIP3 signaling or only noise on the membrane. The latter are mainly very small cells that do not even exhibit transient spot dynamics, as shown in Figure 1E. Although this is a limitation of the parameter Δα, as those two types of cells are indistinguishable, they are only a very few cells whose PIP3 signaling is dominated by noise, and therefore statistically negligible (see Figure 4B, α_min ≃ α_adh ≃ 0.6; Figure 4C, α_min ≃ 0.6). Despite this limitation using Δα as scaling parameter, the major advantage is the normalization of the signaling strength relative to the contact perimeter, since there is a wide distribution of signal activity observed (Figure 4C).

Figures 4D–F show the relation between the scaling parameter Δα and the membrane geometry. We found that larger cells show an oscillatory signaling activity (Δα < 0) on the ventral membrane, while smaller cells show a broad distribution of noise dominant signaling activity in the range of Δα>0 (Figure 4D). Further, smaller cells (cyan color) show a linear correlation between Δα and the ventral membrane area A_adh, i.e., larger A_adh led to larger Δα, and thus to noise-dominant signaling activity on the ventral membrane (Figure 4E). A similar relation was found for cells with negative Δα, where larger cells (red color) had larger A_adh compared to smaller cells V~0.5 pL. The presence of two branches implies a pitchfork-like bifurcation in Δα at Δα ≃ 0. This means that different cells with the same A_adh can take either a smaller value of Δα or larger value. This can be explained by the local membrane curvature at the contact perimeter. Cells of similar ventral membrane area, for example Aadh≃60 μm2, but different volumes (compare small cyan and large red data points) have different contact angles Ω, an indirect measure of the local membrane curvature at the contact perimeter. The inset of Figure 4E shows cells with different membrane curvature (see Discussion for more details). This finding suggests that the local membrane curvature at the contact perimeter regulates the diffusibility between the ventral and non-adherent part of the membrane. Considering the dimensionless adhesion area ratio of r_a = A_adh/A_mem, a linear relationship between r_a and Δα is found (dashed line, Figure 4F). Larger r_a lead to larger positive Δα. The corresponding bifurcation found for A_adh at Δα~0 corresponds to r_a ≃ 10%.

Similar to the geometry of the cells, we have compared the PIP3-enriched domain velocities (see Materials and Methods) to the signal fluctuations. Despite the differences in shape and size of the cell membranes, the average velocity was found to be about 〈v〉~10 μm/min suggesting comparable diffusion properties among all cells. Therefore, no dependence on membrane geometry was observed (Figure 4G). The mean horizontal velocity 〈v_φ〉 shows a similar constant profile, and is the dominant fraction of 〈v〉 (Figure 4H). Contrarily, the mean vertical velocity 〈v_θ〉 is by a factor of about two slower than 〈v_φ〉, and linearly correlated with Δα, i.e., an increase of 〈v_φ〉 is accompanied by an increase in cell size. Figure 4I shows Δα as a function of the mean velocity ratio 〈v_a〉 = 〈v_φ/v_θ〉. A linear relationship is observed similar to the one in Figure 4F.

We conclude that the contact perimeter plays an important role for the pattern organization of the PIP3 signaling dynamics on the membrane. Using weighted polynomial fits, two relationships between the cell geometry (Figure 4F) and the PIP3 domain speed (Figure 4I) were found to be correlated linearly with the introduced contact perimeter dependent parameter Δα (Figure 4E). While it was shown before that the cell asymmetry induces directed domain propagation along the cell periphery (Hörning and Shibata, 2019), this result identifies the contact perimeter as an additional factor that controls the PIP3 signaling organization.

Thus, we summarize that not only the adhesion area ratio r_a but also the PIP3 domain velocity ratio v_a is linearly correlated to the scaling parameter Δα. From those two relationships (dashed lines, Figures 4F,I) we can derive the linear relation between r_a and v_a following the general linear equation v_a = m × r_a + n₀, where m defines the slope and n₀ the intersection with the y-axis at r_a = 0. Not surprisingly, we get the intersection at n₀ ~ 1, since a perfect spherical cell (r_a = 0) will lead to a velocity ratio of unity (Supplementary Figure 4), as shown in numerical simulations before (Hörning and Shibata, 2019). In order to get a more intuitive view of the contact perimeter influence (Figures 2A,B), r_a is transformed to the contact angle Ω from the basic relationship

where R is the radius of the sphere. Equation 7 can be numerically solved as

A good approximation for small contact angles (Ω ≃ 0) is Ω=-sin-1(2ra). Here, Ω is obtained purely based on geometric cell features, but does not account for minor membrane variations and possible constraints based on the mechanical properties of the membrane. Figure 5A shows the relation between Ω and 〈v_a〉. The dashed line denotes the weighted polynomial fit derived from Figures 4F,I. At Ω = 0 we obtain v_a ≃ 1, as the theory predicts when considering isotropic diffusion on membranes. Contrarily the unweighted fit of the data (solid line), where cells with large A_adh (cyan) and large cells with comparable A_adh (red) are underrepresented (Figure 4B, left panel), leads to misleading statistics.

To visualize the influence of the contact perimeter on the signaling dynamics, we utilized the Morlet wavelet transformation (Harang et al., 2012) at all spatially extracted positions on the membrane, as done before only for the ventral membrane of Dictyostelium cells (Taniguchi et al., 2013). Figure 5B shows a scheme of the phase extraction of the entire membrane from the raw data (left side) to the resulting phase map (right side). The phase is adjusted to the propagation front of the domain, so that the red color (= 2π) can be considered as wave front. Applied to the entire spatiotemporal raw data, it is also possible to generate kymographs and snapshots of the time-series, as illustrated in Figures 5C–E. The example illustrates the dynamics of vertical waves of an almost spherical cell (see Figure 2E). The advantage of this visualization method is that the PIP3 wave dynamics and phase singularities can be visualized clearly without application of spatial filters. Figure 5D shows the wave directional change from clockwise to anticlockwise rotation during the 10th and 15th min (Figure 5C) by rotation around a phase singularity (red asterisks), which we denote as a pinning site. The PIP3 wave propagates after the turn around the pinning site toward the center of ventral membrane along an elongated phase line. While the pinning site remains spatially anchored at the contact perimeter until t = 1, 020s (Figure 5E), there is no spatially anchored pinning site at the upper part of the membrane. That means that the PIP3 wave is guided along the contact perimeter, illustrated by the positional change of the phase singularity (red asterisks), because on the upper membrane is no strongly curved membrane as at the contact perimeter. Between t = 1, 020s and t = 1, 060s the phase singularity moves along the contact perimeter and a new pinning site is formed (last panel). Here, we observe two pinning sites, because of the topology constraints of spiral waves in excitable media (Davidsen et al., 2004). While one pinning site changes its position with time on the upper part of the membrane, the pining site on the contact perimeter does not change its position. This topology constraint together with the refractory behavior of the excitable signaling leads to an 〈v_a〉 ~ 1.7, as marked in Figure 5A. Larger cells with large A_adh show an even more dominant influence of the contact perimeter on the PIP3 dynamics (Figure 6). Pinning sites do not only anchor to the contact perimeter (Figure 6A), but can also serve as wave guidance for the wave front (Figure 6B). In this case the PIP3 wave edge aligns with the contact perimeter and leads to even larger speed ratios by forcing the PIP3 wave to propagate parallel to the cell perimeter, in this example to almost 〈v_a〉 ~ 3 (Figure 5A).