Table 1.
Parameters used in the simulations
Figure 1.
Electric field and electroosmotic velocity in the cylindrical model of the polarized and non-polarized cells.
Results of the computer simulation of electric potential, electric field and electroosmotic flow velocity in the axisymmetric model of the cell (cylinder of 10 µm radius and 10 µm height, nucleus–sphere of 3 µm radius, located in the center of the cylinder). Mathematical model is described by eqs. (1–6); parameters are presented in Table 1. Figure 1A represents electric potential (color map) and electric field strength (arrows) in the model of the polarized cell; inward current enters through the top of the cylinder and leaves the cell through the bottom of the cylinder (eq. 2). Electric potential is in the range −0.007÷−0.059 V; strong component of electric field parallel to the cylindrical surface is present. For comparison, Figure 1B presents electric potential and field in the non-polarized cell (zero inward and outward current); here the component of electric field parallel to the surface is practically absent. Figure 1C demonstrates electroosmotic flow circulation (color represents magnitude of the velocity, arrows represent direction) caused by electric field distribution in the polarized cell (Fig. 1A), maximum value of fluid velocity 120 µm/s is in the vicinity of the cylindrical wall, while the average magnitude of the fluid velocity across the cytoplasm is 20 µm/s. Fluid circulates, moving downwards near the cylindrical wall and upwards near the nucleus. For comparison, Figure 1D demonstrates that electroosmotic flow is practically absent (except the filleted corners) in the model of a non-polarized cylindrical cell, where electric field parallel to the surface is absent.
Figure 2.
Evolution of the concentration of messenger proteins initially positioned near the upper corner of the cell.
Results of the computer simulation of the evolution of concentration (color map) and flux (arrows) of the negatively charged messenger proteins (charge = −2e) initially located near the upper corner of the cell. Mathematical model is described by eqs. (7–11). Figure 2A presents the distribution of the messenger proteins 0.01 s after initiation. At t = 0.1 s, as seen at Fig. 2 B, proteins drift with the electroosmotic flow along the cylindrical surface, while at t = 0.5 (Fig. 2C) they are carried by the electroosmotic flow towards the nucleus. The strong flux of messenger proteins to the surface of the nucleus is formed. For comparison, Fig. 2 D demonstrates what happens with messenger proteins in case of the absence of electroosmotic flow, e.g. when zeta-potential, ζ = 0. In this case, negatively charged proteins migrate in the electric field to the least negative part of the cytoplasm, i.e. upper corner of the cell, from where only the small fraction of messengers can reach the nucleus by diffusion against the electric field.
Figure 3.
Evolution of the concentration of messenger proteins initially positioned at the bottom of the cell.
Mathematical model is the same as in Figure 2 except the initial condition that is determined by eq. (12). Figure 3A presents the distribution of the messenger proteins 0.01 s after initiation. Figure 3B demonstrates that at t = 0.1 s a lot of messenger proteins reach the nucleus due to both electroosmotic flow and migration caused by the electric field. Figure 3C illustrates that at t = 0.3 s the layer of messenger proteins is formed in the vicinity of the nucleus. Figure 3D (ζ = 0) demonstrates that the transport is much slower by electromigration relative to electromigration combined with electroosmosis (Fig. 3B).
Figure 4.
Time dependence of the flux of the messenger proteins onto the nucleus.
The flux of the messenger proteins normal to the surface of the nucleus is integrated over this surface. Red curves present the total flux due to electroosmosis, electromigration, and diffusion. For comparison, blue curves present the flux due to pure diffusion, and green curves present the flux due to diffusion and electromigration. In Figure 4A messengers are initially located near the upper corner of the cell (as in Fig. 2), in Figure 4B near the center of the cylindrical surface, in Figure 4C at the bottom of the cylinder (as in Fig. 3), and in Figure 4D at the top of the cylinder. Note that everywhere, except Fig. 4D, electroosmosis facilitates faster and more intense transport of messengers onto the surface of the nucleus. See text for more detailed discussion.
Figure 5.
The 3D model of the fluorescence recovery after photobleaching in the polarized cell.
Mathematical model described by eqs. (7–8) with initial condition (13), representing photobleaching by laser beam. Small cylinder near the top of the cell represents the area across which the recovery of photobleaching is observed (averaged). Figure 5A represents the magnitude of electroosmotic flow velocity (similar to Fig. 1C, but in 3D). Figure 5B presents the concentration of the fluorescently labeled protein after photobleaching, t = 0. Figure 5C–recovery of the fluorescence (concentration of the fluorescent protein) due to pure diffusion, t = 0.05. Figure 5D–recovery of the fluorescence due to electroosmosis and diffusion.
Figure 6.
Dynamics of the recovery of the fluorescence after photobleaching.
Concentration of the fluorescently labeled protein averaged over the small cylinder near the top of the cell presented in Fig. 5. Blue, red, and magenta curves represent recovery of the fluorescence due to pure diffusion for central (blue), biased b = 5 µm (red), and biased b = 7 µm (magenta) locations of the bleaching spot. In case of pure diffusion the time of 50% recovery does not depend on the position of the bleaching spot. On the contrary, in case of electroosmosis and diffusion, the recovery curves differ substantially for the different locations of the bleach spot: central-green, b = 5 µm -cyan, b = 7 µm-yellow. The 50% recovery time differs for these locations: 0.12 s, 0.058 s, and 0.04 s. Therefore diffusion coefficients determined from these values of recovery time might differ 3-fold as well. See text for more detailed discussion.