Research ArticleSYSTEMS BIOLOGY

Traveling and standing waves mediate pattern formation in cellular protrusions

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Science Advances  07 Aug 2020:
Vol. 6, no. 32, eaay7682
DOI: 10.1126/sciadv.aay7682
  • Fig. 1 Transformation of a traveling to a standing wave.

    (A) Kymograph of a wave that traveled, stopped, and extinguished (c2 = 4.2 in inhibitor equation). Dashed arrows indicate where the stopping occurred. (B) Time profiles of a to d from the kymograph in (A), plotting the Laplacian evolution at each of these spatial points. (C) Illustration of how the nullclines are altered by the Laplacian term. The three situations correspond to the time instants marked in (B). The white circle denotes a bifurcation point; the equilibrium is stable if the inhibitor nullcline (red) is to the right of this. The black circle denotes the state, with the immediate trajectory shown by the dashed arrow. (D to F) Example of a standing wave. The panels are as in (A) to (C) but for a wave that transformed into a standing wave on stopping (c2 = 3.9 in inhibitor equation).

  • Fig. 2 Pattern formation and stability.

    (A) (Left) Example of a traveling-to-standing transformation on a longer time scale. The pattern formation is indicated using the variables m and n that show equal spacing of standing branches on a periodic domain. (Right) Zoomed-in version of the activity in the white dashed box. (B) Time evolution of the activity in the red dashed space in (A), showing through the vertical lines the time taken to travel and stop versus the time taken to form the final pattern. (C) Example of deterministic (top) and stochastic (bottom) simulations, where noise (sigma) in the latter causes the standing branches to fall off. (D) Nullclines illustrating the falling off of the stable state to return to the original equilibrium (light red nullcline). (E) Average of 40 simulations with different levels of noise (sigma) and system threshold, which is controlled by the slope of the inhibitor nullcline (c2). A lower slope corresponds to a lower threshold and vice versa.

  • Fig. 3 Traveling and standing phenotypes in cell migration.

    (A and B) Kymograph of PHcrac signaling marker for a latrunculin-treated wild-type (A) and PTEN-null (B) Dictyostelium cells. Images of the cells are shown on the right, with the white circle marked to follow activity at a small region. (C) Wild-type (wt) and PTEN-null cell morphology, with LimE-RFP. Scale bar, 5 μm. (D) PTEN-null example showing actin (left) and signaling (right) markers. Scale bar, 25 μm. (E) Actin dynamics in PTEN-null cells. Arrows indicate small actin rings. This panel is taken from (28) with permission. (F) F-actin wave pattern (GFP-LimE) phenotype induced by RasCQ62L expression in PTEN-null cells (scale bar, 5 μm) forming a pancake-type cell. This panel is taken from (12) with permission.

  • Fig. 4 Simulations of the excitable system recreating experimentally observed wave and morphological phenotypes.

    (A) (Left) Kymographs of normal amoeboid-type protrusions. The yellow dashed line indicates the traveling wave. (Right) Level-set simulations from the activity in (A). (B) (Left) Kymographs of a PTEN-null type protrusion, showing significantly longer thin fingers of activity. The yellow dashed line has much lower slope than that of (A), indicative of the slow velocity of a standing wave. (Right) Level-set simulations from the activity in (B), showing elongated protrusions. (C) Quantification of the duration of activity obtained through simulations from parameter sets of (A) and (B). Patches that covered between 5 and 25% of the domain size were quantified. P value obtained from t test for 180 protrusions. (D) Two-dimensional deterministic simulations manually triggering a wave at the center of the domain to study the time evolution of spatial activity. (E) Images and kymograph showing actin activity in transformed MCF-10A cells. Traveling waves are seen in the images (white arrow) and in the kymograph (dashed circle). Scale bar, 50 μm. (F) Quantification of wave durations seen in transformed MCF-10A cells (three cells). Each point corresponds to a protrusion. The points in the dashed circle indicate those that persisted longer than traveling waves typically do. (G) Images and corresponding kymographs showing PH-AKT activity in a transformed MCF-10A cell. Activity persists at a location (dashed circle) without spreading for over 100 min. Scale bar, 21 μm. (H) Similar standing activity from stochastic two-dimensional simulations.

  • Fig. 5 An all-encompassing protrusion template.

    (A) Phase diagram showing different wave phenotypes through colors and wave ranges through shades. The letters correspond to the particular wave phenotypes. (B) Categorizing wave phenotype thresholds based on wave range, i.e., the fraction of simulation domain occupied by the wave. a, amoeboid; b and c, puncta/little waves; d, PTEN-null; e, pancake; f, oscillator.

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