Полежаев А.А. Физический Институт им. П.Н. Лебедева

Математические модели и численные методы в биоматематике МЕХАНИЗМ ФОРМИРОВАНИЯ ЭЛЕКТРО-ХИМИЧЕСКИХ СТРУКТУР НА ПОВЕРХНОСТИ КЛЕТКИ ВОДОРОСЛИ CHARA(MECHANISM OF ELECTRO-CHEMICAL PATTERN FORMATION ON THE SURFACE OF THE ALGAE Chara CELL) Полежаев А.А. Физический Институт им. П.Н. Лебедева 11 января 2011 г.

Outline • Introduction • Patterns on the membrane of the algae Chara cell: experiment • 1D model. Conditions for diffusion-convection instability • 2D case. Dependence on the geometry • Conclusion

Light-triggered pH bands in Chara cells Stationary pH distribution around the cell. The pattern was visualized using Phenol Red at a concentration of 25 μM. Dark spots correspond to alkaline regions.

Profile measured with glass-insulated antimony pH microelectrodes with tip diameter of 10-20 mm

Banding pH profiles measured at different light intensities: 1 - 100 W/m2, 2 - 5 W/m2, 3 - 2.5 W/m2, 4 - 250 mW/m2.

Formation of banding patterns upon illumination of a dark-adapted Chara cell: (a) initial pH profile under stationary illumination; (b) suppression of banding after 100 min in darkness; (c) appearance of pH fluctuations after 5 min in light; (d) pH pattern with pronounced bands - 20 min in light; (e) disappearance of one pronounced band - 50 min in light. The initial pH profile (dotted line) is shown for comparison in (d).

pH pattern induced by local illumination of the central region of Chara internode and evolution of banding pattern after transition from local to overall illumination. (a) Two closely positioned bands form at the borders of the illuminated zone; the bar under the curve indicates the cell fragment with illuminated (width 4 mm) and darkened regions; (b)-(d) Banding profiles measured at different times after transition from local to overall illumination: (b) 12 min (c) 25 min and (d) 50 min. New bands appear sequentially. One of two initial bands in the doublet is eventually eliminated.

Scheme of the Chara cell membrane P+ diffusion out Cell membrane proton pump ATP in CO2 P+ photosynthesis light

Model protons outside proton pump passive transport diffusion CO2 inside photosyntesis diffusion trough the membrane diffusion ATP inside photosyntesis diffusion ATP production waste

Reduced model If

Turing (diffusion) instability Conditions for the instability of the uniform state (0,0) • ad-bc >0, • a+d<0, • d<0, • a>0, e)

Cytoplasmic flow (cyclosis) V vacuole V

Differential flow instability Rovinsky, Menzinger, 1992 Bulychev et al., 2001 where and

Differential flow instability The uniform stationary state becomes unstable if

Reaction-diffusion-convection type model with symmetric flows

Linear analysis

Linear analysis One of the roots of this equation is positive, if let us denote then

Conditions for the break of stability of the uniform state (0, 0, 0) due to diffusion and/or symmetric convective flows ad-bc >0, a+d<0, d<0, a>0, where

Numerical simulations (1D case)

V=0 D=0.4

V=0 D=0.3

V=0.5 D=0.5

V=1 D=0.5

V=3.5 D=1.5

2D case where y H/2 V 0 L x V -H/2

L=100 H=10

L=100 H=15

Conclusion • Cross-current convective flows of one of the variables (namely, corresponding to the inhibitor) can effectively increase the inhibitor transport, thus leading to instability. • Peculiarities of the emerging patterns, depending on the geometry of the integration region, namely, the cell length and diameter, were studied numerically. It appeared that stationary patterns are formed in the case of sufficiently narrow region (small cell diameter), what corresponds to the actual cell size. Computations have shown that if the thickness of the cell were larger, then instead of stationary acid and alkaline zones there would appear patterns, drifting in the opposite directions.

Collaboration • S. C. MüllerOtto-von-Guericke-Universität, Magdeburg, Germany • A. A. Bulychev, G. YU. RiznichenkoMoscow State University, Russia

Полежаев А.А. Физический Институт им. П.Н. Лебедева

Полежаев А.А. Физический Институт им. П.Н. Лебедева

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