Phase geometries of two-dimensional excitable waves govern self-organized morphodynamics of amoeboid cells

Abstract

In both randomly moving Dictyostelium and mammalian cells, phosphatidylinositol (3,4,5)-trisphosphate and F-actin are known to propagate as waves at the membrane and act to push out the protruding edge. To date, however, the relationship between the wave geometry and the patterns of amoeboid shape change remains elusive. Here, by using phase map analysis, we show that morphology dynamics of randomly moving Dictyostelium discoideum cells can be characterized by the number, topology, and position of spatial phase singularities, i.e., points that represent organizing centers of rotating waves. A single isolated singularity near the cellular edge induced a rotational protrusion, whereas a pair of singularities supported a symmetric extension. These singularities appeared by strong phase resetting due to de novo nucleation at the back of preexisting waves. Analysis of a theoretical model indicated excitability of the system that is governed by positive feedback from phosphatidylinositol (3,4,5)-trisphosphate to PI3-kinase activation, and we showed experimentally that this requires F-actin. Furthermore, by incorporating membrane deformation into the model, we demonstrated that geometries of competing waves explain most of the observed semiperiodic changes in amoeboid morphology.

Fig. 1.

Inhibition of actin polymerization or PI3Kinase activity reduces the rate of wave nucleation and simplifies the wave patterns. (A) PIP3 (PHcrac-RFP) and F-actin (LimEΔCoil-GFP) waves at the basal membrane. (B and C) Wave nucleation rate depended on F-actin and PI3Kinase. Wave nucleation was suppressed in cells treated with latrunculin A (B) and PI3Kinase inhibitor LY294002 (C) in a dose-dependent manner. (D–G) Wave patterns in pharmacologically treated cells and a pirA⁻ cell. (D) Transient waves were observed in cells treated with 60 μM LY294002. Rotational (E) and reflecting (F) waves were observed in cells treated with 5 μM latrunculin. (G) Sporadic localized accumulation of PIP3 in a pirA⁻ cell. Red and green colors indicate fluorescence intensity of PH-cracRFP and PTEN-GFP, respectively. (Scale bars: 5 μm.)

Fig. 2.

Phase map analyses of PIP3 waves in latrunculin-treated cells. (A) Phase mapping of a single rotating wave. Phase of oscillating PIP3 at every spatial point was extracted from a grayscale image of fluorescence intensity (Upper Left) by Morlet wavelet transformation (25). (Lower Left) A representative time series from a fixed position. (Center) Amplitude (Upper) and phase (Lower) of the wavelet transform. The frequency component having the maximum magnitude at each time point was obtained by ridge extraction (white line), and the phase was calculated accordingly. (Right) White solid circle depicts a phase singularity with +1 charge that corresponds to clockwise rotation. (Scale bar, 10 μm.) (B–D) Codistribution of phases of PIP3, F-actin, and PTEN waves: ϕ_PIP3, ϕ_actin, and ϕ_PTEN, respectively. Data in B–D were obtained from Movie S1 A, C, and D respectively. (E) Time series of phase maps during spiral reversal in a cell treated with 3 μM latrunculin. A core of a clockwise rotating spiral wave disappeared at t = 126 s. A new wave that appeared at t = 252 s created a pair of phase singularities (t = 318 s). One of the singularities reached the border and disappeared, resulting in a dominance of a left-handed spiral. (Scale bar, 10 μm.)

Fig. 3.

Phase map analyses of PIP3 waves in nontreated cells. (A) Phase analysis of PIP3 wave in a freely moving cell. Symmetric expansion of a circular wavefront was supported by phase singularities of +1 and −1 charge located at both ends (t = 54–84 and 348–396 s). Rotational PIP3 waves were maintained by an isolated phase singularity (t = 180–222 s). The arrowheads indicate a front–front interaction that resulted in wave merger and extinction (t = 54 and 264 s). The dotted regions indicate front–back interactions of the wave, which create a pair of phase singularities (t = 0, 222, and 294 s). (B) Front–front interaction: Wavefronts (yellow) simply merged and annihilated each other after the collision. (C) Front–back interaction: Waves did not merge but instead created a pair of phase singularities. This occurred for nucleation at the back of a wave or when a wavefront (yellow) caught up with a waveback (blue). (D) Phase change upon birth of singularities was discontinuous, indicating a strong phase resetting of type 0.

Fig. 4.

Numerical simulations of phosphoinositide waves and membrane deformation. Concentrations of PIP3 and PIP2 in the simulations (white background panels) are colored in red and green, respectively. Experimental data (black background) shown for comparison are fluorescence intensity of PHcracRFP (red) and PTEN-GFP (green). (A–D) Simulations in a fixed circular domain. (A) For weak feedback strength, waves were extinguished at the boundary so that they propagated only transiently as observed in LY294002-treated cells (Fig. 1D). For moderate strength of feedback, the system showed either spiral (B) or reflecting (C) waves, similar to those observed in cells treated with latrunculin (Fig. 1 E and F). For exceedingly large feedback strength, the system exhibited no excitability and only small patches of transient PIP3 enrichment similar to those in the pirA⁻ mutant (Fig. 1G) were observed (D). The strength of the feedback is increased by lowering the value of K_K (SI Text). K_K = 6.0 (A), 5.7 (B), 4.2 (C), and 2.5 (D). (E) Phase diagram of the cell behavior. Parameter β indicates the strength of the PTEN-mediated dephosphorylation reaction. Red represents no excitability; yellow, reflection at the edge; and blue, extinction at the edge. Spirals are observed in the shaded region. Symbols represent parameters in A–D. (F–I) Simulation results with a deforming boundary. Concentric (F), translational (G), and rotational (H) wave propagation induced deformation of distinct forms at the cell edge. Head-to-head collision of waves induced a lateral membrane extension. (Scale bars: 10 μm.) Detailed descriptions of the model and parameters are provided in SI Text and Table S1.